Benjamin's  Machine  Design. 

By  CHARLES  H.  BENJAMIN,  Professor 
in  the  Case  School  of  Science. 

Hoskins'  Hydraulics. 

By  L.  M.  HOSKINS,  Professor  in  Leland 
Stanford  University. 

Adams'  Alternating-current 
Machines. 

By  C.  A.  ADAMS,  Professor  in  Harvard 
University.      [In  preparation.  ] 


HENRY   HOLT  AND   COMPANY 

NEW  YORK  CHICAGO 


MACHINE   DESIGN 


BY 


CHARLES    H.    BENJAMIN 

Professor  of  Mechanical  Engineering  in  the 
Case  School  of  Applied  Science 


NEW  YORK 
HENRY    HOLT    AND    COMPANY 

1908 


Copyright,  1906 

BY 

HENRY  HOLT  AND  COMPANY 


PREFACE. 


THIS  book  embodies  to  a  considerable  extent  the 
writers  experience  in  teaching  and  in  commercial 
work.  While  the  underlying  mechanical  principles  of 
machine  design  are  permanent,  the  application  of  them 
is  continually  changing.  The  researches  of  the  ex- 
perimenter and  the  practice  of  the  builder  are  always 
showing  good  reasons  for  the  modification  of  design* 

Although  the  present  work  was  prepared  primarily 
for  a  text-book,  it  contains  mainly  what  the  writer  has 
found  necessary  in  his  own  practice  as  an  engineer. 
As  far  as  possible  the  formulas  for  the  strength  and 
stiffness  of  machine  details  have  been  fortified  by  the 
results  of  experiments  or  by  the  practical  experience  of 
manufacturers* 

Attention  is  called  particularly  to  the  experiments 
on  cast-iron  cylinders,  pipe  fittings,  helical  springs, 
roller  bearings,  gear  teeth,  pulley  arms,  and  the 
bursting  strength  of  fly-wheels. 

What  the  student  needs  to  learn  before  graduation 
is  also  what  he  needs  to  remember  after,  and  it  is 
hoped  that  this  book  contains  the  necessary  facts  and 
principles  and  not  too  much  else. 

C.  H.  B. 


iii 

361349 


TABLE  OF  CONTENTS. 


I.  UNITS  AND  TABLES 1 

I.  Units.     2.   Abbreviations.     3.    Materials.    4.    Notation. 
5.  Formulas.     6.  Profiles  of  uniform  strength.     7.  Factors 
of  safety. 

II.  FRAME  DESIGN 15 

8.  General  principles  of  design.  9.  Machine  supports.  10. 
Machine  frames. 

III.  CYLINDERS  AND  PIPES 25 

II.  Thin  shells.     12.  Thick  shells.     13.  Steel  and  wrought 
iron  pipe.     14.  Strength  of  boiler  tubes.     15.  Pipe  fittings. 
16.  Steam  cylinders.     17.  Thickness  of  flat  plates. 

IV.  FASTENINGS  54 

18.  Bolts  and  nuts.     19.  Machine    screws.     20.  Eye    bolts 
and  hooks.     21.  Riveted  joints.     22.  Lap  joints.     23.  Butt 
joints  with  two  straps.     24.  Efficiency  of  joints.     25.  Butt 
joints  with  unequal  straps.     26.  Practical  rules.     27.  Riv- 
eted joints  for  narrow  plates.     28.  Joint  pins.     29.  Cotters. 

V.  SPRINGS , 73 

30.  Helical  springs.  31.  Square  wire.  32.  Experiments. 
33.  Springs  in  torsion.  34.  Flat  springs.  35.  Elliptic  and 
semi-elliptic  springs. 

VI.  SLIDING  BEARINGS 86 

36.  Slides  in  general.  37.  Angular  slides.  38.  Gibbed 
slides.  39.  Flat  slides.  40.  Circular  guides.  41.  Stuffing 
boxes. 

VII.  JOURNALS,  PIVOTS  AND  BEARINGS 96 

42.  Journals,     43.  Adjustment.    44.  Lubrication.    45.  Fric- 
tion of  journals.     46.  Limits  of  pressure.     47.  Heating  of 
journals.     48.  Experiments.     49.  Strength  and  stiffness  of 
journals.     50.  Caps     and     bolts.     51.  Step     bearings.     52. 
Friction  of  pivots.     53.  Flat  collar.     54.  Conical  pivot.     55. 
Schiele's  pivot.     56.  Multiple  bearing. 

V 


vi  MACHINE  DESIGN. 

CHAPTER.  PAGE. 

VIII.  BALL  AND  ROLLER  BEARINGS 118 

57.  General     principles.     58.  Journal     bearings.     59.  Step 
bearings.     60.  Materials  and  wear.     61.  Design  of  bearings. 

62.  Roller  bearings.  63.  Grant  roller  bearing.  64.  Hyatt 
rollers.  65.  Roller  step  bearings. 

IX.  SHAFTING,  COUPLINGS  AND  HANGERS 130 

'  66.  Strength     of     shafting.     67.  Couplings.     68.  Clutches. 

69.  Coupling  bolts.  70.  Shafting  keys.  71.  Hangers  and 
boxes. 

X.  GEARS,  PULLEYS  AND  CRANKS 146 

72.  Gear  teeth.  73.  Strength  of  teeth.  74.  Lewis'  formula. 
75.  Experimental  data.  76.  Teeth  of  bevel  gears.  77. 
Rim  and  arms.  78.  Sprocket  wheels  and  chains,  79. 
Silent  chains.  80.  Cranks  and  levers. 

XI.  FLY-WHEELS 166 

81.  In  general.  82.  Safe  speed  for  wheels.  83.  Experiments 
on  fly  wheels.  84.  Wooden  pulleys.  85.  Rims  of  cast-iron 
gears.  86.  Rotating  discs.  87.  Plain  discs.  88.  Conical 
discs.  89.  Discs  with  logarithmic  profile.  90.  Bursting 
speeds. 

XII.  TRANSMISSION  BY  BELTS  AND  ROPES 184 

91.  Friction     of     belting.     92.  Strength    of     belting.     93. 
Taylor's  experiments.     94.  Rules  for  width  of  belts.    95. 
Speed     of     belting.     96.  Manila     rope     transmission.     97. 
Strength  of  Manila  ropes.    98.  Wire  rope  transmission. 


TABLES. 


PAGE. 

I.    STRENGTH  OF  WROUGHT  METALS 6 

II.    STRENGTH  OP  CAST  METALS 7 

III.  VALUES  OP  Q  IN  COLUMN  FORMULA 10 

Ilia.    VALUES  OF  S  AND  K  IN  COLUMN  FORMULA 10 

IV.  CONSTANTS  OF  CROSS-SECTION 11 

V.    FORMULAS  FOR  LOADED  BEAMS. 12 

VI.    SIZES  OF  IRON  AND  STEEL  PIPE 31 

VII.    SIZES  OF  EXTRA  STRONG  PIPE 33 

VIII.    SIZES  OF  DOUBLE  EXTRA  STRONG  PIPE   34 

IX.    SIZES  OF  IRON  AND  STEEL  BOILER  TUBES 36 

X.    STRENGTH  OF  STANDARD  SCREWED  PIPE  FITTINGS 41 

XI.    BURSTING  STRENGTH  OF  CAST  IRON  CYLINDERS 45 

XII.    STRENGTH  OF  REINFORCED  CYLINDERS 47 

XIII.  STRENGTH  OF  CAST  IRON  PLATES 52 

XIV.  STRENGTH  OF  IRON  OR  STEEL  BOLTS 54 

XV.    DIMENSIONS  OF  MACHINE  SCREWS 57 

XVI.    DIMENSIONS  OF  RIVETED  LAP  JOINTS 65 

XVII.    DIMENSIONS  OF  RIVETED  BUTT  JOINTS 65 

XVIII.    STRENGTH  AND  STIFFNESS  OF  HELICAL  SPRINGS 77 

XIX.    FRICTION  OF  PISTON  ROD  PACKINGS 93 

XX.  "          "        "        "  "         94 

XXI.  "          "        "        "  "         94 

XXII.    FRICTION  OF  JOURNAL  BEARING 108 

XXIII.  FRICTION  OF  ROLLER  AND  PLAIN  BEARINGS 126 

XXIV.  "          "        "          "        "  "         127 

XXV.    DIAMETERS  OF  SHAFTING c 132 

XXVI.    PROPORTIONS  OF  GEAR  TEETH 148 

XXVII.    SIZES  OF  TEST  FLY-WHEELS 172 

XXVIII.    FLANGES  AND  BOLTS  OF  TEST  FLY-WHEELS. 172 

XXIX.    FAILURE  OF  FLANGED  JOINTS 173 

XXX.    SIZES  OF  LINKED  JOINTS 173 

XXXI.    FAILURE  OF  LINKED  JOINTS , 174 

XXXII.    BURSTING  SPEEDS  OF  ROTATING  Discs , 182 

XXXIII.  HORSE  POWER  OF  MANILA  ROPE 192 

XXXIV.  HORSE  POWER  OF  WIRE  ROPE 194 

vii 


MACHINE  DESIGN. 


CHAPTER  I. 

UNITS  AND   TABLES. 

1.  Units.     In  this  book  the  following  units  will  be 
used  unless  otherwise  stated. 

Dimensions  in  inches. 

Forces  in  pounds. 

Stresses  in  pounds  per  square  inch. 

Velocities  in  feet  per  second. 

Work  and  energy  in  foot  pounds. 

Moments  in  pounds  inches. 

Speeds  of  rotation  in  revolutions  per  minute. 

The  word  stress  will  be  used  to  denote  the  resistance 
of  material  to  distortion  per  unit  of  sectional  area. 
The  word  deformation  will  be  used  to  denote  the  dis- 
tortion of  a  piece  per  unit  of  length.  The  word  set  will 
be  used  to  denote  total  permanent  distortion  of  a  piece. 

In  making  calculations  the  use  of  the  slide-rule  and 
of  four-place  logarithms  is  recommended  ;  accuracy  is 
expected  only  to  three  significant  figures. 

2.  Abbreviations.     The  following  abbreviations  are 
among  those  recommended  by  a  committee  of  the 
American  Society  of  Mechanical  Engineers  in  Decem- 
ber, 1904?  an$  will  be  used  throughout  the  book, 


MACHINE  DESIGN. 


NAME. 

ABBREVIATION. 

Inches        

.    in. 

Feet   

.    ft. 

Yards         

.       .    yd. 

Miles.        ...                 . 

.     spell  out. 

.    Ib. 

Tons           

.     spell  out. 

Seconds      

.     sec. 

Minutes      ...... 

.    min. 

Hours         ...... 

.     hr. 

Linear        ...... 

.     lin. 

Square       ...... 

.     sq. 

cu. 

Per             

.    spell  out. 

Fahrenheit         ..... 

.     fahr. 

Percentage         ..... 

.     %  or  per  cent. 

Brake  horse  power     .... 

.     b.h.p. 

Electric  horse  power 

.     e.h.p. 

Indicated  horse  power 

.     i.h.p. 

British  thermal  units 

.     B.t.u. 

Diameter 

.     Diam. 

3.  Materials.  The  principal  materials  used  in 
machine  construction  are  given  in  the  following  tables 
with  the  physical  characteristics  of  each. 

By  wrought  iron  is  meant  commercially  pure  iron 
which  has  been  made  from  molten  pig-iron  by  the 
puddling  process  and  then  squeezed  and  rolled,  thus 
developing  the  fiber.  This  iron  has  been  largely  sup- 
planted by  soft  steel. 

Ordinary  wrought  iron  contains  from  0.1%  to  0.3% 
of  carbon.  Soft  steel  may  contain  no  more  than  this, 
but  is  different  in  structure.  The  particles  of  iron  in 
the  puddling  process  are  more  or  less  enveloped  in 
slag  or  earthy  matter  and  as  the  bloom  is  squeezed  and 


MATERIALS.  3 

rolled  the  particles  become  fibers  separated  from  each 
other  by  a  thin  sheath  or  covering  of  slag,  and  it  is 
this  that  gives  such  iron  its  characteristic  structure. 
The  principal  impurities  in  the  iron  are  phosphorus 
from  the  ore  and  sulphur  from  the  fuel. 

In  making  steel,  on  the  other  hand,  the  molten 
iron  has  had  the  silicon  and  carbon  removed  by  a  hot 
blast,  either  passing  through  the  liquid  as  in  the  Bes- 
semer converter,  or  over  its  surface  as  in  the  open- 
hearth  furnace.  A  suitable  quantity  of  carbon  and 
manganese  has  then  been  added  and  the  metal  poured 
into  ingot  molds.  If  the  steel  is  then  reheated  and 
passed  through  a  series  of  rolls,  structural  steel  and 
rods  or  rails  result. 

Bessemer  steel  contains  from  0.1%  to  0.6%  of  carbon 
and  has  a  fine  granular  structure.  This  material  has 
been  much  used  for  rails. 

Open  hearth  steel  differs  from  Bessemer  but  little  in 
its  chemical  composition  but  is  usually  more  reliable  in 
quality  on  account  of  the  more  deliberate  nature  of 
the  process  of  manufacture.  It  is  generally  used  for 
boiler  plate  and  for  steel  castings.  Two  grades  of 
boiler  plate  are  commonly  known  as  marine  steel  and 
flange  steel,  the  latter  being  of  the  better  quality. 

Steel  castings  are  poured  directly  from  the  open 
hearth  furnace  and  allowed  to  cool  without  any  draw- 
ing or  rolling.  They  are  coarser  and  more  crystalline 
than  the  rolled  steel. 

Crucible  steel  usually  contains  from  one  to  one  and 
a  half  per  cent  of  carbon,  is  relatively  high  priced  and 
only  used  for  cutting  tools.  It  is  made  by  melting 
steel  in  an  air-tight  crucible  with  the  proper  additions 
of  carbon  and  manganese. 

Cast  iron  is  made  directly  from  the  pig  by  remelt- 


4:  MACHINE  DESIGN. 

ing  and  casting,  is  granular  in  texture  and  contains 
from  two  to  five  per  cent,  of  carbon.  A  portion  of  the 
carbon  is  chemically  combined  with  the  iron  while  the 
remainder  exists  in  the  form  of  graphite.  The  harder 
and  whiter  the  iron  the  more  carbon  is  found  chemi- 
cally combined.  Silicon  is  an.  important  element  in 
cast  iron  and  influences  the  rate  of  cooling.  The  more 
slowly  iron  cools  after  melting  the  more  graphite  forms 
and  the  softer  the  iron. 

Two  per  cent  of  silicon  gives  a  soft  gray  iron  of  a 
high  tensile  strength. 

Machinery  iron  contains  usually  from  one  and  one 
half  to  two  per  cent  of  silicon. 

Malleable  iron  is  cast  iron  annealed  and  partially  de- 
carbonized by  being  heated  in  an  annealing  oven  in  con- 
tact with  some  oxidizing  material  such  as  haematite  ore. 
This  process  makes  the  iron  tougher  and  less  brittle. 

All  castings  including  those  made  from  alloys  are 
somewhat  unreliable  on  account  of  hidden  flaws  and  of 
the  strains  developed  by  shrinkage  while  cooling. 

The  so-called  high-speed  or  air -hardening  tool  steels 
are  alloys  of  steel  with  various  substances  such  as 
chromium  (chrome  steel),  tungsten  (Mushet  steel), 
molybdenum,  etc.,  etc. 

They  are  characterized  by  extreme  hardness  at 
comparatively  high  temperatures.  Their  other  physi- 
cal characteristics  are  not  of  particular  interest. 

The  addition  of  nickel  to  steel  increases  its  ultimate 
strength  and  also  raises  its  elastic  limit.  The  tensile 
strength  is  sometimes  as  high  as  200,000  Ib.  per  sq.  in.  and 
the  steel  is  also  tough  and  well  adapted  to  resist  shocks. 

The  bronzes  are  alloys  of  copper  and  tin,  copper  and 
zinc,  or  of  all  three.  The  copper-tin  alloys  usually 
contain  85  or  90  per  cent  of  copper  and  are  expensive. 


MATERIALS.  5 

The  copper-zinc  alloys,  or  brasses  as  they  are  sometimes 
called,  should  have  from  60  to  70  per  cent  of  copper 
for  maximum  strength  and  ductility. 

Bronzes  high  in  tin  and  low  in  copper  are  weak,  but 
have  considerable  ductility  and  make  good  metals  for 
bearings.  Tin  80,  copper  10  and  antimony  10  is  Babbitt 
metal,  so  much  used  to  line  journal  bearings,  the 
antimony  increasing  the  hardness. 

The  late  Dr.  Thurston's  experiments  on  the  copper- 
tin-zinc  alloys  showed  a  maximum  strength  for  copper 
55,  zinc  43  and  tin  2  per  cent.  The  tensile  strength  of 
this  mixture  was  nearly  70,000  Ib.  per  sq.  in. 

Phosphor  bronze  is  a  copper  alloy  with  a  small 
amount  of  phosphorus  added  to  prevent  oxidation  of 
the  copper  and  thereby  strengthen  the  alloy. 

Manganese  bronze  is  an  alloy  of  copper  and  man- 
ganese, usually  containing  iron  and  sometimes  tin.  A 
bronze  containing  about  84  per  cent  copper,  14  per  cent 
manganese  and  a  little  iron,  has  much  the  same  physical 
characteristics  as  soft  steel  and  resists  corrosion  much 
better. 

The  constants  for  strength  and  elasticity  given  in 
the  tables  are  only  fair  average  values,  and  should  be 
determined  for  any  special  material  by  direct  experi- 
ment when  it  is  practicable.  Many  of  the  constants  are 
not  given  in  the  table  on  account  of  the  lack  of  reliable 
data  for  their  determination. 

The  strength  of  steel,  either  rolled  or  cast,  depends 
so  much  upon  the  percentages  of  carbon,  phosphorus 
and  manganese,  that  any  general  figures  are  liable  to 
be  misleading.  Structural  steel  usually  has  a  tensile 
strength  of  about  65,000  Ib.  per  sq.  in.,  while  boiler 
plate  usually  has  less  carbon,  a  low  tensile  strength 
and  good  ductility. 


MACHINE  DESIGN. 


03  >>      J 

~  •&       H 

r'fl 


8 


H 


bO 


a 


02 


CO        1C        O 


»a      o      o 

«O        OO        Oi 


o 


O 


•a 


O*         -2  _4 


to 


02 


MATERIALS. 


g 


w 


o  o 

o  o 

o  o 

CO  00 


8 


8  MACHINE  DESIGN. 

4.  Notation. 

Arc  of  contact  =0  radians. 

Area  of  section  =A  sq.  in. 

Breadth  of  section  =b  in. 

Coefficient  of  friction  =/ 

Deflection  of  beam  =A  in. 

Depth  of  section  =h  in. 

Diameter  of  circular  section  —d  in. 

Distance  of  neutral  axis  from  outer  fiber  —y  in. 

Elasticity,  modulus  of, 

in  tension  and  compression  =E 

in  shearing  and  torsion  =  Gr 

Heaviness,  weight  per  cu.  ft.  —w 

Length  of  any  member  =  I  in. 

Load  or  dead  weight  =  W  Ib. 

Moment,  in  bending  =Jflb.-in. 

in  twisting  =  Tib. -in. 

Moment  of  inertia 

rectangular  =1 

polar  =  J 

Pitch  of  teeth,  rivets,  etc.  =p  in. 

Radius  of  gyration  =r  in. 

Section  modulus,  bending 

ij 

twisting 

Stress  per  unit  of  area  =S 

Velocity  =v  ft  per  sec. 

5.  Formulas. 

Simple  Stress. 
Tension,  compression  or  shear,   £=— r (0 


NOTATION.  9 

Bending  under  Transverse  Load. 

orr 

General  equation,  M  =  — (2) 

Eectangular  section,  M= — ^—    .     .     .     .(3) 

Rectangular  section,  bh2=—~- (4) 

Circular  section,  M=^—?     .     .     .     .(5) 

Circular  section,  d=    r®'^     .  .    .     .(6) 

«  •  'V      >S 

Torsion  or  Twisting. 
General  equation, 
Circular  section, 
Circular  section, 
Hollow  circular  section, 
Other  values  of  -    and  -  may  be  taken  from  Table  4. 

y        y 

Combined  Bending  and  Twisting. 
Calculate  shaft  for  a  twisting  moment, 

Column  subject  to  Bending. 
Use  Rankine's  formula,  ~^=~    ~^ (12) 


The  values  of  r2  may  be  taken  from  Table  IV.  The 
subjoined  table  gives  the  average  values  of  g,  while  S 
is  the  compressi ve  strength  of  the  material. 


10 


MACHINE  DESIGN. 


TABLE  III. 

VALUES  OF  q  IN  FORMULA  12. 


Material. 

Both 
ends 
fixed. 

Fixed 
and 
round. 

Both 
ends 
round. 

Fixed 
and 
free. 

Timber 

1 

1.78 

4 

16 

Cast  Iron  

3000 
1 

3000 
1.78 

3000 
4 

3000 
16 

Wrought  Iron  

5000 
1 

5000 
1.78 

5000 
4 

5000 
16 

Steel  

36000 
1 

36000 
1.78 

36000 
4 

36000 
16 

25000 

25000 

25000 

25000 

Carnegie's  hand-book  gives  S  =  50000  for  medium 
steel  columns  and  g=inrTRnr,  rcfonr  and  inm  for  the 
three  first  columns  in  above  table. 

In  this  formula,  as  in  all  such,  the  values  of  the 
constant  should  be  determined  for  the  material  used  by 
direct  experiment  if  possible. 

Or  use  straight  line  formula,  --r=  S  —  Jc-    .    .    .(12a) 

.1  / 

TABLE  Ilia. 

VALUES  OF  S  AND  k  IN  FORMULA  (12a). 

(Merriman's  Mechanics  of  Materials.) 


Kind  of  Column. 

S 

k 

Limit  - 

Wrought  Iron  : 
Flat  ends  

42000 

128 

218 

Hinged  ends.  .  .  ."  

42000 

157 

178 

Round  ends 

42000 

203 

138 

Mild  Steel  : 
Flat  ends  

52500 

179 

195 

Hinged  ends  

52500 

220 

159 

Round  ends.. 

52500 

284 

123 

Cast  Iron  : 
Flat  ends 

80000 

438 

122 

Hinged  ends  

80000 

537 

99 

Round  ends 

80000 

693 

77 

Oak  : 
Flat  ends  

5400 

28 

128 

FORMULAS. 


11 


Carnegie's    hand-book    gives    allowable    stress   for 
structural  columns  of  mild  steel  as  12000  for  lengths 

less   than   90   radii,    and   as    17100—57-   for    longer 

columns. 
This  allows  a  factor  of  safety  of  about  four. 

TABLE  IV. 

CONSTANTS  OF  CROSS-SECTION. 


Form  of 
Section  and 
Area  A. 

Square  of 
Radius  of 
Gyration 

Moment 
of 
Inertia 

Section 
Modulus 
I 

y 

Polar 
Moment 
of  Inertia 
J 

Torsion 
Modulus 
J 

y 

Rectangle 

/i2 

bh3 

bh* 

bh*+b*h 

W+Vh 

bh 

12 

12 

6 

12 

6  V62+/i2 

Square 

d2 

d4 

d3 

d4 

d3 

d2 

12 

12 

6 

6 

4T24 

Hollow 

Rectangle 

bhz—bji3l 

bh3—blh3l 

67i3—  fW, 

or  7-beam 

I2(bh  b  h  ) 

12 

Qh 

bh-blhl 

Circle 

d2 

TTd4 

d3 

TTd4 

d3 

5*. 

16 

64 

10.2 

32 

5.1 

Hollow 

Circle 

*+*, 

7r(d4—  d4j) 

10.  2d* 

7r(d4-d4,) 

d4—  d4! 

64 

32 

5.  Id 

Ellipse 

a2 

*W 

6«2 

7r(ba3+  ab3} 

&«*+«&' 

f* 

16 

64 

10.2 

64 

10.2a 

Values  of  /  and  J  for  more  complicated  sections  can  be  worked 
out  from  those  in  table. 


12 


MACHINE  DESIGN. 


TABLE  V. 

FORMULAS  FOR  LOADED  BEAMS. 


Beams  of  Uniform  Cross-section. 

Maximum. 
Moment. 
M 

Maximum. 
Deflection. 

A 

Wl 

Wl3 

Cantilever,  uniform  load  

Wl 

3EI 
WP 

Simple  beam,  load  at  middle  

2 
Wl 

8EI 
Wl3 

Simple  beam,  uniform  load  

4 
Wl 

4SEI 
5W73 

Beam  fixed  at  one  end,  supported  at  other, 
load  at  middle  

8 

3WI 
~W 

384#7 
.Q182W13 

Beam  fixed  at  one  end,  supported  at  other, 
uniform  load 

Wl 
g 

.0054TFZ3 

Beam  fixed  at  both  ends,  load  at  middle  .  . 

Beam  fixed  at  both  ends,  uniform  load.  .  .  . 
Beam  fixed  at  both  ends,  load  at  one  end, 

Wl 

T 

Wl 

12 

Wl 

~2~ 

Wl3 
1S2EI 

Wl* 

m±EI 

Wl* 
12EI 

6.  Profiles  of  Uniform  Strength.  In  a  bracket  or 
beam  of  uniform  cross-section  the  stress  on  the  outer 
row  of  fibers  increases  as  the  bending  moment  increases 
and  the  piece  is  most  liable  to  break  at  the  point  where 
the  moment  is  a  maximum.  This  difficulty  can  be 
remedied  by  varying  the  cross-section  in  such  a  way 
as  to  keep  the  fiber  stress  constant  along  the  top  or 
bottom  of  the  piece.  The  following  table  shows  the 
shapes  to  be  used  under  different  conditions. 


FACTORS  OF  SAFETY. 


13 


Type. 

Load. 

Plan. 

Elevation. 

Cantilever. 

Center  .... 

Rectangle  .  . 

Parabola,  axis  horizontal. 

Cantilever.  . 
Simp.  Beam 

Uniform  .  .  . 
Center 

Rectangle  .  . 
Rectangle  . 

Triangle. 
Two    parabolas    intersecting 

Simp.  Beam 

Uniform  .  .  . 

Rectangle  .  . 

under  load. 
Ellipse,  major  axis  horizontal. 

The  material  is  best  economized  by  maintaining  a 
constant  breadth  and  varying  the  depth  as  indicated. 

This  method  of  design  is  applicable  to  cast  pieces 
rather  than  to  those  that  are  forged  or  cut. 

The  maximum  deflection  of  cantilevers  and  beams 
having  a  profile  of  uniform  strength  is  greater  than 
when  the  cross-section  is  uniform,  fifty  per  cent, 
greater  if  the  breadth  varies,  and  one  hundred  percent 
greater  if  the  depth  varies. 

7.  Factors  of  Safety.  A  factor  of  safety  is  the  ratio 
of  the  ultimate  strength  of  any  member  to  the  ordinary 
working  load  which  will  come  upon  it.  This  factor  is 
intended  to  allow  for  :  (a)  Overloading  either  inten- 
tional or  accidental,  (b)  Sudden  blows  or  shocks. 

(c)  Gradual    fatigue    or    deterioration    of     material. 

(d)  Flaws  or  imperfections  in  the  material. 

To  a  certain  extent  the  term  "  factor  of  ignorance" 
is  justifiable  since  allowance  is  made  for  the  unknown. 
Certain  fixed  laws  may  guide  one,  however,  in  making 
the  selection  of  a  factor.  It  is  a  well-known  fact  that 
loads  in  excess  of  the  elastic  limit  are  liable  to  cause 
failure  in  time.  Therefore,  when  the  elastic  limit  of 
the  material  can  be  determined,  it  should  be  used  as  a 
basis  rather  than  to  use  the  ultimate  strength. 


14  MACHINE  DESIGN. 

Furthermore,  suddenly  applied  loads  will  cause 
about  double  the  stress  due  to  dead  loads.  These  two 
considerations  point  to  four  as  the  least  factor  that 
should  be  used  when  the  ultimate  strength  is  taken  as 
a  basis.  Pieces  subject  to  stress  alternately  in  opposite 
directions  should  have  large  factors  of  safety. 

The  following  table  shows  the  factors  used  in  good 
practice  under  various  conditions  : 

Structural  steel  in  buildings  .  .  4 

"  "     "  bridges  .  .  5 

Steel  in  machine  construction         .  .  6 

"      "  engine  "  10 

Steel  plate  in  boilers  ...  5 

Cast  iron  in  machines          .  .  .  6  to  15 

Castings  of  bronze  or  steel  should  have  larger  factors 
than  rolled  or  forged  metal  on  account  of  the  possibility 
of  flaws. 

Cast  iron  should  not  be  used  in  pieces  subject  to 
tension  or  bending  if  there  is  a  liability  of  shocks  or 
blows  coming  on  the  piece. 


CHAPTER  II. 

FRAME  DESIGN. 

8.  General  Principles  of  Design.  The  working  or 
moving  parts  should  be  designed  first  and  the  frame 
adapted  to  them. 

The  moving  parts  can  be  first  arranged  to  give  the 
motions  and  velocities  desired,  special  attention  being 
paid  to  compactness  and  to  the  convenience  of  the 
operator. 

Novel  and  complicated  mechanisms  should  be 
avoided  and  the  more  simple  and  well  tried  devices 
used. 

Any  device  which  is  new  should  be  first  tried  in  a 
working  model  before  being  introduced  in  the  design. 

The  dimensions  of  the  working  parts  for  strength 
and  stiffness  must  next  be  determined  and  the  design 
for  the  frame  completed.  This  may  involve  some 
modification  of  the  moving  parts. 

In  designing  any  part  of  the  machine,  the  metal 
must  be  put  in  the  line  of  stress  and  bending  avoided 
as  far  as  possible. 

Straight  lines  should  be  used  for  the  outlines  of 
pieces  exposed  to  tension  or  compression,  circular 
cross  sections  for  all  parts  in  torsion,  and  profile  of 
uniform  fiber  stress  for  pieces  subjected  to  bending 
action. 

Superfluous  metal  must  be  avoided  and  this  excludes 
all  ornamentation  as  such.  There  should  be  a  good 

15 


16  .  .  MACHINE  DESIGN. 

practical  reason  for  every  pound  of  metal  in  the 
machine. 

An  excess  of  metal  is  sometimes  needed  to  give 
inertia  and  solidity  and  prevent  vibration,  as  in  the 
frames  of  machines  having  reciprocating  parts,  like 
engines,  planers,  slotting  machines,  etc. 

Mr.  Oberlin  Smith  has  characterized  this  as  the 
"  anvil "  style  of  design  in  contradistinction  to  the 
" fiddle"  style. 

In  one  the  designer  relies  on  the  mass  of  the  metal, 
in  the  other  on  the  distribution  of  the  metal,  to  resist 
the  applied  forces. 

A  comparison  of  the  massive  Tangye  bed  of  some 
large  high-speed  engines  with  the  comparatively  slight 
girder  frame  used  in  most  Corliss  engines,  will  em- 
phasize this  difference. 

It  may  be  sometimes  necessary  to  waste  metal  in 
order  to  save  labor  in  finishing,  and  in  general  the  aim 
should  be  to  economize  labor  rather  than  stock. 

The  designers  should  be  familiar  with  all  the  shop 
processes  as  well  as  the  principles  of  strength  and 
stability.  The  usual  tendency  in  design,  especially  of 
cast  iron  work,  is  towards  unnecessary  weight. 

All  corners  should  be  rounded  for  the  comfort  and 
convenience  of  the  operator,  no  cracks  or  sharp  inter- 
nal angles  left  where  dirt  and  grease  may  accumulate, 
and  in  general  special  attention  should  be  paid  to  so 
designing  the  machine  that  it  may  be  safely  and  con- 
veniently operated,  that  it  may  be  easily  kept  clean, 
and  that  oil  holes  are  readily  accessible.  The  ap- 
pearance of  a  machine  in  use  is  a  key  to  its  working 
condition. 

Polished  metal  should  be  avoided  on  account  of  its 
tendency  to  rust,  and  neither  varnish  nor  bright  colors 


FIG.  1. 


OLD  PLANING  MACHINE.     AN  EXAMPLE  OF  ELABORATE 
OHNAM  ENTATION. 


GENERAL  PRINCIPLES  OF  DESIGN.  17 

tolerated.  The  paint  should  be  of  some  neutral  tint 
and  have  a  dead  finish  so  as  not  to  show  scratches  or 
dirt. 

Beauty  is  an  element  of  machine  design,  but  it  can 
only  be  attained  by  legitimate  means  which  are  appro- 
propriate  to  the  material  and  the  surroundings. 

Beauty  is  a  natural  result  of  correct  mechanical 
construction  but  should  never  be  made  the  object  of 
design. 

Harmony  of  design  may  be  secured  by  adopting  one 
type  of  cross-section  and  adhering  to  it  throughout, 
never  combining  cored  or  box  sections  with  ribbed 
sections.  In  cast  pieces  the  thickness  of  metal  should 
be  uniform  to  avoid  coolin'g  strains,  and  for  the  same 
reason  sharp  corners  should  be  absent.  The  lines  of 
crystallization  in  castings  are  normal  to  the  cooled 
surface  and  where  two  flat  pieces  come  together  at 
right  angles,  the  interference  of  the  two  sets  of  crys- 
tals forms  a  plane  of  weakness  at  the  corner.  This  is 
best  obviated  by  joining  the  two  planes  with  abend  or 
sweep. 

Rounding  the  external  corner  and  filleting  the  in- 
ternal one  is  usually  sufficient.  Where  two  parts  come 
together  in  such  a  way  as  to  cause  an  increase  of 
thickness  of  the  metal  there  are  apt  to  be  "  blow  holes  " 
or  "hot  spots"  at  the  junction  due  to  the  uneven 
cooling. 

"  Strengthening  "  flanges  when  of  improper  propor- 
tions or  in  the  wrong  location  are  frequently  a  source 
of  weakness  rather  than  strength.  A  cast  rib  or  flange 
on  the  tension  side  of  a  plate  exposed  to  bending,  will 
sometimes  cause  rupture  by  cracking  on  the  outer  edge. 
When  apertures  are  cut  in  a  frame  either  for  core- 
prints  or  for  lightness,  the  hole  or  aperture  should  be 


18  MACHINE  DESIGN. 

the  symmetrical  figure,  and  not  the  metal  that  sur- 
rounds it,  to  make  the  design  pleasing  to  the  eye. 

The  design  should  be  in  harmony  with  the  material 
used  and  not  imitation.  For  example,  to  imitate 
structural  work  either  of  wood  or  iron  in  a  cast-iron 
frame  is  silly  and  meaningless. 

Machine  design  has  been  a  process  of  evolution. 
The  earlier  types  of  machines  were  built  before  the 
general  introduction  of  cast-iron  frames  and  had 
frames  made  of  wood  or  stone,  paneled,  carved  and 
decorated  as  in  cabinet  or  architectural  designs. 

When  cast  iron  frames  and  supports  were  first 
introduced  they  were  made  to  imitate  wood  and  stone 
construction,  so  that  in  the  Earlier  forms  we  find  panels, 
moldings,  gothic  traceries  and  elaborate  decorations  of 
vines,  fruit  and  flowers,  the  whole  covered  with  con- 
trasting colors  of  paint  and  varnished  as  carefully  as  a 
piece  of  furniture  for  the  drawing-room.  Relics  of 
this  transition  period  in  machine  architecture  may  be 
seen  in  almost  every  shop.  One  man  has  gone  down 
to  posterity  as  actually  advertising  an  upright  drill 
designed  in  pure  Tuscan. 

9.  Machine  Supports.  The  fewer  the  number  of 
supports  the  better.  Heavy  frames,  as  of  large  en- 
gines, lathes,  planers,  etc.,  are  best  made  so  as  to  rest 
directly  on  a  masonry  foundation.  Short  frames  as 
those  of  shapers,  screw  machines  and  milling  machines, 
should  have  one  support  of  the  cabinet  form.  The  use 
of  a  cabinet  at  one  end  and  legs  at  the  other  is  offensive 
to  the  eye,  being  inharmonious.  If  two  cabinets  are 
used  provision  should  be  made  for  a  cradle  or  pivot  at 
one  end  to  prevent  twisting  of  the  frame  by  an  uneven 
foundation.  The  use  of  intermediate  supports  is 


GENERAL  PRINCIPLES  OF  DESIGN.  19 

always  to  be  condemned,  as  it  tends  to  make  the  frame 
conform  to  the  inequalities  of  the  floor  or  foundation 
on  what  has  been  aptly  termed  the  "  caterpillar  prin- 
ciple." 

A  distinction  must  be  made  between  cabinets  or 
supports  which  are  broad  at  the  base  and  intended  to 
be  fastened  to  the  foundation,  and  legs  similar  to  those 
of  a  table  or  chair.  The  latter  are  intended  to  simply 
rest  on  the  floor,  should  be  firmly  fastened  to  the 
machine  and  should  be  larger  at  the  upper  end  where 
the  greatest  bending  moment  will  come. 

The  use  of  legs  instead  of  cabinets  is  an  assumption 
that  the  frame  is  stiff  enough  to  withstand  all  stresses 
that  come  upon  it,  unaided  by  the  foundation,  and  if 
that  is  the  case  intermediate  supports  are  unne- 
cessary. 

Whether  legs  or  cabinets  are  best  adapted  to  a  cer- 
tain machine  the  designer  must  determine  for  himself. 

Where  two  supports  or  pairs  of  legs  are  necessary 
under  a  frame,  it  is  best  to  have  them  set  a  certain 
distance  from  the  ends,  and  make  the  overhanging 
part  of  the  frame  of  a  parabolic  form,  as  this  divides 
up  the  bending  moment  and  allows  less  deflection  at 
the  center.  Trussing  a  long  cast-iron  frame  with  iron 
or  steel  rods  is  objectionable  on  account  of  the  differ- 
ence in  expansion  of  the  two  metals  and  the  liability 
of  the  tension  nuts  being  tampered  with  by  work- 
men. 

The  sprawling  double  curved  leg  which  originated 
in  the  time  of  Louis  XIV  and  which  has  served  in  turn 
for  chairs,  pianos,  stoves  and  finally  for  engine  lathes 
is  wrong  both  from  a  practical  and  aesthetic  stand- 
point. It  is  incorrect  in  principle  and  is  therefore 
ugly. 


20  MACHINE  DESIGN. 

EXERCISE. 

1. — Apply  the  foregoing  principles  in  making  a  written 
criticism  of  some  engine  or  machine  frame  and  its  supports. 

(a)  Girder  frame  of  engine. 

(b)  Tangye  bed  of  air  compresser. 

(c)  Bed,  uprights  and  supports  of  iron  planing  machine. 

(d)  Bed  and  supports  of  engine  lathe. 

(e)  Cabinet  of  shaping  or  milling  machine. 

(f)  Frame  of  upright  drill. 

10.  Machine  Frames.  For  general  principles  of 
frame  design  the  reader  is  referred  to  Chapter  2.  Cast 
iron  is  the  material  most  used  but  steel  castings  are 
now  becoming  common  in  situations  where  the  stresses 
are  unusually  great,  as  in  the  frames  of  presses,  shears 
and  rolls  for  shaping  steel. 

Cored  vs.  Rib  Sections.  Formerly  the  flanged  or 
rib  section  was  used  almost  exclusively,  as  but  a  few 
castings  were  made  from  each  pattern  and  the  cost  of 
the  latter  was  a  considerable  item.  Of  late  years  the 
use  of  hollow  sections  has  become  more  common  ;  the 
patterns  are  more  durable  and  more  easily  molded 
than  those  having  many  projections  and  the  frames 
when  finished  are  more  pleasing  in  appearance. 

The  first  cost  of  a  pattern  for  hollow  work,  including 
the  cost  of  the  core-box,  is  sometimes  considerably 
more  but  the  pattern  is  less  likely  to  change  its  shape 
and  in  these  days  of  many  castings  from  one  pattern, 
this  latter  point  is  of  more  importance.  Finally  it 
may  be  said  that  hollow  sections  are  usually  stronger 
for  the  same  weight  of  metal  than  any  that  can  be 
shaped  from  webs  and  flanges. 


MACHINE  FRAMES. 


21 


Resistance  to  Bending.  Most  machine  frames  are 
exposed  to  bending  in  one  or  two 
directions.  If  the  section  is  to  be 
ribbed  it  should  be  of  the  form 
shown  in  Fig.  3.  The  metal 
being  of  nearly  uniform  thickness 
and  the  flange  which  is  in  tension 
having  an  area  three  or  four  times 
that  of  the  compression  flange. 
In  a  steel  casting  these  may  be 
more  nearly  equal.  The  hollow 


Fig.  3. 


0w/y//MMW//t/ 

\ 

% 

; 

; 

p 

In 

B 


Fig.  4. 


section  may  be  of  the  shape  shown  in  Fig.  4,  a  hollow 
rectangle  with  the  tension  side  re-enforced  and  slightly 
thicker  than  the  other  three 
sides.  The  re-enforcing  flanges 
at  A  and  B  may  often  be  utilized 
for  the  attaching  of  other  mem- 
bers to  the  frame  as  in  shapers 
or  drill  presses.  The  box  section 
has  one  great  advantage  over 
the  I  section  in  that  its 
moment  of  resistance  to  side 
bending  or  to  twisting  is  usually  much  greater.  The 
double  I  or  the  TJ  section  is  common  where  it  is 
necessary  to  have  two  parallel 
ways  for  sliding  pieces  as  in  lathes 
and  planers.  As  is  shown  in  Fig. 
5  the  two  Is  are  usually  connected 
at  intervals  by  cross  girts. 

Besides  making  the  cross-section 
of  the  most  economical  form,  it  is 
often    desirable  to    have   such   a 
longitudinal  profile  as  shall  give  a 
Fig.  5.  uniform  fiber  stress  from  end  to 


22  MACHINE  DESIGN. 

end.  This  necessitates  a  parabolic  or  elliptic  outline 
of  which  the  best  instance  is  the  housing  or  upright  of 
a  modern  iron  planer. 

A  series  of  experiments  made  in  1902  under  the 
direction  of  the  author,  on  the  modulus  of  rupture  of 
cast  iron  beams  of  the  same  weight  but  different  cross- 
sections  gave  interesting  results.  Beginning  with  the 
solid  circular  section,  which  failed  under  a  transverse 
load  of  7,500  lb.,  square,  rectangular,  hollow  and 
/  shaped  sections  were  tested  until  a  maximum  was 
reached  in  the  /  section  with  heavy  tension  flange 
which  broke  under  a  load  of  38,000  lb.  Channel  and 
T  shaped  sections  such  as  are  appropriate  for  fly-wheel 
rims  were  also  tested  with  the  ribs  in  tension  and  in 
compression. 

The  strength  of  such  sections  was  found  to  be  from 
two  to  three  times  as  great  when  the  ribs  were  in  com- 
pression as  when  they  were  in  tension. 

Resistance  to  Twisting.  The 
hollow  circular  section  is  the  ideal 
form  for  all  frames  or  machine 
members  which  are  subjected  to 
torsion.  If  subjected  also  to 
bending  the  section  may  be  made 
elliptical  or,  as  is  more  common, 
thickened  on  two  sides  by  making 
the  core  oval.  See  Fig.  6.  As 
Fig.  6.  has  already  been  pointed  out  the 

box  sections  are  in  general  better 

adapted  to  resist  twisting  than  the  ribbed  or  /  sec- 
tions. 

Frames  of  Machine  Tools.  The  beds  of  lathes  are 
subjected  to  bending  on  account  of  their  own  weight 
and  that  of  the  saddle  and  on  account  of  the  downward 


MACHINE  FRAMES.  23 

pressure  on  the  tool  when  work  is  being  turned.  They 
are  usually  subjected  to  torsion  on  account  of  the  un- 
even pressure  of  the  supports.  The  box  section  is  then 
the  best ;  the  double  I  commonly  used  is  very  weak 
against  twisting.  The  same  principle  would  apply  in 
designing  the  beds  of  planers  but  the  usual  method  of 
driving  the  table  by  means  of  a  gear  and  rack  prevents 
the  use  of  the  box  section.  The  uprights  of  planers 
and  the  cross  rail  are  subjected  to  severe  bending 
moments  and  should  have  profiles  of  uniform  strength. 
The  uprights  are  also  subject  to  side  bending  when  the 
tool  is  taking  a  heavy  side  cut  near  the  top.  To  pro- 
vide for  this  the  uprights  may  be  of  a  box  section  or 
may  be  reinforced  by  outside  ribs. 

The  upright  of  a  drill  press  or  vertical  shaper  is 
exposed  to  a  constant  bending  moment  equal  to  the 
upward  pressure  on  the 
cutter  multiplied  by  the 
distance  from  center  of 
cutter  to  center  of  up- 
right.    It  should  then  be 
of  constant  cross-section 
from  the  bottom  to  the 
top  of  the  straight  part. 

The    curved    or    goose-        ^ 

necked    portion    should  Fig  7 

then  taper  gradually. 

The  frame  of  a  shear  press  or  punch  is  usually  of 
the  Gr  shape  in  profile  with  the  inner  fibers  in  tension 
and  the  outer  in  compression.  The  cross-section  should 
be  as  in  Fig.  3  or  Fig.  4,  preferably  the  latter,  and 
should  be  graduated  to  the  magnitude  of  the  bending 
moment  at  each  point.  (See  Fig.  7.) 


24  MACHINE  DESIGN. 

EXERCISES. 

1.  Discuss  the  stresses  and  the  arrangement  of  material  in 
the  girder  frame  of  a  Corliss  engine. 

2.  Ditto  in  the  G  frame  of  a  band  saw. 

PROBLEM. 

Design  a  G  frame  similar  to  that  shown  in  Fig.  7,  for  a  shear 
press  capable  of  shearing  a  bar  of  mild  steel  1|  by  1£  inches 
and  having  a  gap  four  inches  high  and  twenty-six  inches  deep. 


CHAPTEE  III. 


Fig.  8. 


CYLINDER  AND   PIPES. 

ii.  Thin  Shells.  Let  Fig.  8  represent  a  section  of 
a  thin  shell,  like  a  boiler 
shell,  exposed  to  an  inter- 
nal pressure  of  p  pounds 
per  sq.  inch.  Then,  if  we 
consider  any  diameter  AB, 
the  total  upward  pressure  A\ 
on  upper  half  of  the  shell 
will  balance  the  total  down- 
ward pressure  on  the  lower 
half  and  tend  to  separate 
the  shell  at  A  and  B  by 
tension. 

Let  d= diameter  of  shell  in  inches. 

r= radius  of  shell  in  inches. 
1= length  of  shell  in  inches. 
t= thickness  of  shell  in  inches. 
S=  tensile  strength  of  material. 

Draw  the  radial  line  CP  to  represent  the  pressure 
on  the  element  P  of  the  surface. 

Area  of  element  at  P=lrdO. 

Total  pressure  on  element  =plrd&. 

Vertical  pressure  on  element  =plr  sin  OdO. 

Total  vertical  pressure  on  APB= ^  C plr  sin  OdO=%plr. 

25 


26  MACHINE  DESIGN. 

The  area  to  resist  tension  at  A  and  B=2tl  and  its 
total  strength  =2tlS. 

Equating  the  pressure  and  the  resistance 


pr    pd 

2  S     4:S 


.         .         .         .     (13) 

The  total  pressure  on  the  end  of  a  closed  cylindrical 
shell  =Trr*p  and  the  resistance  of  the  circular  ring  of 
metal  which  resists  this  pressure  =2irrtS. 

Equating  : 

t 

Therefore  a  shell  is  twice  as  strong  in  this  direction 
as  in  the  other.  Notice  that  this  same  formula  would 
apply  to  spherical  shells. 

In  calculating  the  pressure  due  to  a  head  of  water 
equals  h,  the  following  formula  is  useful : 

p=OAMh.         .         .          .          (15) 

In  this  formula  h  is  in  feet  and  p  in  pounds  per 
square  inch. 

PROBLEMS. 

1.  A  cast-iron  water  pipe  is  12  inches  in  internal  diameter 
and  the  metal  is  .45  inches  thick.     What  would  be  the  factor 
of  safety,  with  an  internal  pressure  due  to  a  head  of  water  of 
250  feet  ? 

2.  What  would  be  the  stress  caused  by  bending  due  to 
weight,  if  the  pipe  in  Ex.  1  were  full  of  water  and  24  feet  long, 
the  ends  being  merely  supported  ? 

3.  A  standard  lap- welded  steam  pipe,  8  inches  in  nominal 
diameter  is  0.32  inches  thick  and  is  tested  with  an  internal 
pressure  of  500  pounds  per  sq.  inch.     What  is  the  bursting 
pressure  and  what  is  the  factor  of  safety  above  the  test  pres- 
sure, assuming  S=  40000  ? 


THICK  SHELLS.  -ft 

12.  Thick  Shells.  There  are  several  formulas  for 
thick  cylinders  and  no  one  of  them  is  entirely  satis- 
factory. It  is  however  generally  admitted  that  the 
tensile  stress  caused  by  internal  pressure  in  such  a 
cylinder  is  greatest  at  the  inner  circumference  and 
diminishes  according  to  some  law  from  there  to  the 
exterior  of  the  shell.  This  law  of  variation  is  expressed 
differently  in  the  different  formulas. 

Barlow's  Formulas.  Here  the  cylinder  diameters 
are  assumed  to  increase  under  the  pressure,  but  in  such 
a  way  that  the  volume  of  metal  remains  constant. 
Experiment  has  proved  that  in  extreme  cases  this  last 
assumption  is  incorrect.  Within  the  limits  of  ordinary 
practice  it  is,  however,  approximately  true. 

Let  di  and  d2  be  the  interior  and  exterior  diameters 

j  _  ,7 
in  inches  and  let  t=   2      *  be  the  thickness  of  metal. 

2i 

Let  I  be  the  length  of  cylinder  in  inches. 

Let  Si  and  S*  be  the  tensile  stresses  in  Ibs.  per  sq. 
inch  at  inner  and  outer  circumferences. 

The  volume  of  the  ring  of  metal  before  the  pressure 
is  applied  will  be  : 


and  if  the  two  diameters  are  assumed  to  increase  th 
amounts  xl  and  xz  under  pressure  the  final  volume 
will  be  : 


Assuming  the  volume  to  remain  the  same  : 
df  -  d?  =  ( 


28  MACHINE  DESIGN. 

Neglecting  the  squares  of  Xi  and  xz  this  reduces  to 


or  the  distortions  are  inversely  as  the  diameters. 
The  unit  deformations  will  be  proportional  to 

and  the  stresses  Si  and  S2  will  be  in  the  same  ratio  : 

j^f          /y*  C$          /~J  ^ 

or  the  stresses  vary  inversely  as  the  squares  of  the 
diameters.  Let  S  be  the  stress  at  any. diameter  d, 
then  : 

S=—^=~~  (where  r  is  radius) 

and  the  total  stress  on  an  element  of  the  area  l.dr  is  : 


Integrating    this    expression    between  the  limits  -^ 

and  -~  for  r  and  multiplying  by  2  we  have  : 

~ 


Eq  uating  this  to  the  pressure  which  tends  to  produce 
rupture,  pdl,  where  p  is  the  internal  unit  pressure, 
there  results  : 


The  formula  (13)  for  thin  shells  gives  P= 

By  comparing  this  with  formula  (16)  it  will  be  seen 
that  in  designing  thick  shells  the  external  diameter 
determines  the  working  pressure  or  : 


THICK  SHELLS. 


Lame's  Formula. — In  this  discussion  each  particle 
of  the  metal  is  supposed  to  be  subjected  to  radial  com- 
pression and  to  tangential  and  longitudinal  tension  and 
to  be  in  equilibrium  under  these  stresses. 

Using  the  same  notation 
as  in  previous  formula  : 


for  the  maximum  stress 
at  the  interior, 

',  •     (18) 


for  the  stress  at  the  outer 
surface. 

Fig.  9  illustrates  the 
variation  in  S  from  inner 
to  outer  surface. 

Solving  for  cZa  in  (17)  we  have 


Fig.  9. 


*-*^Ps 


.(19) 


A  discussion  of  Lame's  formula  may  be  found  in 
most  works  on  strength  of  materials. 

PROBLEMS. 

1.  A  hydraulic  cylinder  has  an  inner  diameter  of  8  inches,  a 
thickness  of  four  inches  and  an  internal  pressure  of  1500  Ibs. 
per  sq.  in.     Determine  the  maximum  stress  on  the  metal  by 
Barlow's  and  Lame's  formulas. 

2.  Design  a  cast-iron  cylinder  6  inches  internal  diameter  to 
carry  a  working  pressure  of  1200  Ibs.  per  sq.  in.  with  a  factor 
of  safety  of  10. 

3.  A  cast-iron  water  pipe  is  1  inch  thick  and  12  inches  in- 
ternal diameter.     Required  head  of  water  which  it  will  carry 
with  a  factor  of  safety  of  6. 


30  MACHINE  DESIGN. 

13.  Steel  and  Wrought  Iron  Pipe.  Pipe  for  the 
transmission  of  steam,  gas  or  water  may  be  made  of 
wrought  iron  or  steel.  Cast-iron  is  used  for  water 
mains  to  a  certain  extent,  but  its  use  for  either  steam 
or  gas  has  been  mostly  abandoned.  The  weight  of 
cast-iron  pipe  and  its  unreliability  forbid  its  use  for 
high  pressure  work. 

Wrought  iron  pipe  up  to  and  including  one  inch  in 
diameter  is  usually  butt-welded,  and  above  that  is  lap- 
welded.  Steel  pipes  may  be  either  welded  or  may  be 
drawn  without  any  seam.  Electric  welding  has  been 
successfully  applied  to  all  kinds  of  steel  tubing,  both 
for  transmitting  fluids  and  for  boiler  tubes. 

The  following  tables  are  taken  by  permission  from 
the  catalogue  of  the  Crane  Company  and  show  the  stand- 
ard dimensions  for  steam  pipe  and  for  boiler  tubes. 

Ordinary  standard  pipe  is  used  for  pressures  not 
exceeding  100  Ib.  per  sq.  in.,  extra  strong  pipe  for  the 
pressures  prevailing  in  steam  plants  where  compound 
and  triple  expansion  engines  are  used,  while  the 
double  extra  is  employed  in  hydraulic  work  under  the 
heavy  pressures  peculiar  to  that  sort  of  transmission. 


STEEL  AND  WROUGHT  IRON  PIPE. 


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Tests  made  by  the  Crane  Company  on  ordinary  com- 
mercial pipe  such  as  is  listed  in  Table  VI  showed  the 
following  pressures  : 

8  in.  diam.          .         .          2000  Ib.  per  sq.  in. 
10         "       .          .  2300         "          " 

12         "       .          .  1500         "          " 

The  pipe  was  not  ruptured  at  these  pressures. 


36 


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38  MACHINE  DESIGN. 

14.  Strength  of  Boiler  Tubes.  When  tubes  are  used 
in  a  so-called  fire-tube  boiler  with  the  gas  inside  and  the 
water  outside,  they  are  exposed  to  a  collapsing  pres- 
sure. 

The  same  is  true  of  the  furnace  flues  of  internally 
fired  boilers.  Such  a  member  is  in  unstable  equilibrium 
and  it  is  difficult  to  predict  just  when  failure  will 
occur. 

Experiments  on  small  wrought-iron  tubes  have 
shown  the  collapsing  pressure  to  be  about  80  per  cent. 
of  the  bursting  pressure.  With  short  tubes  set  in 
tube  sheets  the  length  would  have  considerable  in- 
fluence on  the  strength,  but  ordinary  boiler  tubes 
collapsing  at  the  middle  of  the  length  would  not  be 
influenced  by  the  setting. 

The  strength  of  such  tubes  is  probably  proportional 


-j  )    where  t  is  the  thickness  and  d  the  diameter. 
dj 

D.  K.  Clark  gives  for  large  iron  flues  the  following 
formula  : 

.......  (20) 


where  P  is  the  collapsing  pressure  in  Ib.  per  sq.  in. 
These  flues  had  diameters  varying  from  30  in.  to  50  in. 
and  thickness  of  metal  from  f  in.  to  T7T  in. 

Prof.  K.  T.  Stewart  has  recently  made  some  in- 
teresting experiments  on  the  collapsing  pressure  of 
lap-welded  steel  tubes  and  reported  the  results  to  the 
American  Society  of  Mechanical  Engineers.  (See 
Transactions,  Vol.  XXVTI.) 

It  will  only  be  possible  here  to  give  some  of  the  gen- 
eral conclusions,  as  stated  by  the  author  in  his  paper  : 

1.  The  length  of  tube,  between  transverse  joints 
tending  to  hold  it  to  a  circular  form,  has  no  practical 


STRENGTH  OF  BOILER  TUBES.  39 

influence  upon  the  collapsing  pressure  of  a  commercial 
lap-welded  steel  tube  so  long  as  this  length  is  not  less 
than  about  six  diameters  of  tube. 

2.  The  formulae,  as  based  upon  the  present  research, 
for  the  collapsing  pressure  of  modern  lap- welded  Bes- 
semer steel  tubes,  are  as  follows  : 


P=1000(l-  J  1-1600  ~.)    .     .     .(A) 

P=86670-~-1386 (B) 

Where  P=  collapsing  pressure,  pounds  per  sq.  inch. 

d= outside  diameter  of  tube  in  inches 

£= thickness  of  wall  in  inches 
Formula  (A)  is  for  values  of  P  less  than  581  pounds, 

or  for  values  of  -r-  less  than  0.023,  while  formula  (B) 

is  for  values  greater  than  these. 

These  formulae,  while  strictly  correct  for  tubes  that 
are  20  feet  in  length  between  transverse  joints  tending 
to  hold  them  to  a  circular  form,  are,  at  the  same  time, 
substantially  correct  for  all  lengths  greater  than  about 
six  diameters. 

They  have  been  tested  for  seven  diameters,  ranging 
from  3  to  10  inches,  in  all  obtainable  thicknesses  of 
wall,  and  are  known  to  be  correct  for  this  range. 

3.  The  apparent  fiber  stress  under  which  the  different 
tubes  failed  varied  from  about  7000  pounds  for  the 
relatively  thinnest  to  35000  pounds  per  square  inch  for 
the  relatively  thickest  walls. 

Since  the  average  yield  point  of  the  material  was 
37000  and  the  tensile  strength  58000  pounds  per  square 
inch,  it  would  appear  that  the  strength  of  a  tube 
subjected  to  a  collapsing  fluid  pressure  is  not  dependent 


40  MACHINE  DESIGN. 

alone  upon  either  the  elastic  limit  or  ultimate  strength 
of  the  material  constituting  it. 

15.  Pipe  Fittings.  Steam  pipe  up  to  and  including 
pipe  two  inches  in  diameter  is  usually  equipped  with 
screwed  fittings,  including  ells,  tees,  couplings,  valves, 
etc. 

Pipe  of  a  larger  size,  if  used  for  high  pressures, 
should  be  put  together  with  flanged  fittings  and  bolts. 
One  great  advantage  of  the  latter  system  is  the  fact 
that  a  section  of  pipe  can  easily  be  removed  for  repairs 
or  alterations. 

Small  connections  are  usually  made  of  cast-iron  or 
malleable  iron.  While  the  latter  are  neater  in  ap- 
pearance they  are  more  apt  to  stretch  and  cause  leaky 
joints.  The  larger  fittings  are  made  of  cast-iron  or 
cast-steel.  Such  fittings  can  be  obtained  in  various 
weights  and  thicknesses,  to  correspond  to  those  grades 
of  pipe  listed  in  the  tables. 

The  designer  should  have  at  hand  catalogues  of  pipe 
fittings  from  the  various  manufacturers,  as  these  will 
give  in  detail  the  proportions  of  all  the  different  con- 
nections. 

For  pressures  not  exceeding  100  Ibs.  per  sq.  in.  rubber 
and  asbestos  gaskets  can  be  used  between  the  flanges, 
but  for  higher  pressures  or  for  superheated  steam 
corrugated  metallic  gaskets  are  necessary. 

In  1905  some  very  interesting  experiments  on  the 
strength  of  standard  screwed  elbows  and  tees  were 
made  by  Mr.  S.  M.  Chandler,  a  graduate  of  the  Case 
School,  and  published  by  him  in  "  Power"  for  October, 
1905. 

The  fittings  were  taken  at  random  from  the  stock  of 
the  Pittsburg  Valve  and  Fittings  Co.,  and  three  of 


PIPE  FITTINGS.  41 

each  size  were  tested  to  destruction  by  hydraulic  pres- 
sure. 

The  following  table  gives  a  summary  of  the  results 
obtained.  The  values  which  are  starred  in  the  table 
were  obtained  from  fittings  which  had  purposely  been 
cast  with  the  core  out  of  center  so  as  to  make  one  wall 
thinner  than  the  other.  These  values  are  not  included 
in  the  averages. 

TABLE  X. 

BURSTING  STRENGTH  OF  STANDARD  SCREWED  FITTINGS, 
PRESSURES  IN  POUNDS  PER  SQUARE  INCH. 

SIZE.  ELBOWS.  AVERAGE. 


2* 

3500 

3300 

3400 

3400 

3 

2400 

2600 

2100* 

2500 

8* 

2100 

1700* 

2400 

2250 

4 

2800 

2500 

2500 

2600 

4i 

2000* 

2600 

2600 

2600 

5 

2600 

2500 

2500 

2538   . 

6 

2600 

2200 

2300 

2367 

7 

1800 

2100 

1900* 

1950 

8 

1700 

1600 

1700 

1667 

9 

1800 

1800 

1900 

1833 

10 

1800 

1700 

1600 

1700 

12 

1100 

1200 

900* 

1150 

SIZE. 


TEES. 


AVERAGE. 


li 

3400 

3300 

3300 

3333 

H 

3400 

3200 

2800* 

3300 

2 

2500 

2800 

2500 

2600 

2i 

2400 

2100* 

2500 

2450 

3 

1400* 

1900 

1800 

1850 

3* 

1200* 

1500 

1800 

1650 

4 

1800 

2100 

1700 

1867 

4* 

1100* 

1400 

1400 

1400 

5 

1700 

1300* 

1500 

1600 

6 

1400 

1500 

1100* 

1450 

7 

1400 

1400 

1500 

1433 

8 

1200* 

1400 

1300 

1350 

9 

1300 

1400 

1200 

1300 

10 

1100 

1300 

1200 

1200 

12 

1100 

1000 

1100 

1067 

*  Made  with  eccentric  core. 


42  MACHINE  DESIGN. 

These  tests  show  a  large  apparent  factor  of  safety 
for  any  pressures  to  which  screwed  fittings  are  usually 
subjected. 

The  failure  of  such  fittings  in  practice  must  be  at- 
tributed to  faulty  workmanship  in  erection,  such  as 
screwing  too  tight,  lack  of  allowance  for  expansion 
and  poor  drainage. 

The  average  tensile  strength  of  the  cast-iron  used  in 
the  above  fittings  was  20000  Ibs.  per  sq.  in. 

PROBLEMS. 

1.  Determine  the  bursting  pressure  of  a  wrought  iron  steam 

pipe  6  inches  nominal  diameter. 

(a)  If  of  standard  dimensions. 

(b)  If  extra  strong. 

(c)  If  double  extra  strong. 

2.  Compare  the  above  with  the  strength  of  standard  screwed 

elbows  and  tees  of  the  same  size. 

3.  Determine  the  probable  collapsing  pressure  of  a  charcoal 

iron  boiler-tube  of  two  inches  nominal  diameter. 

16.  Steam  Cylinders.  Cylinders  of  steam  engines 
can  hardly  be  considered  as  coming  under  either  of  the 
preceeding  heads.  On  the  one  hand  the  thickness  of 
metal  is  not  enough  to  insure  rigidity  as  in  hydraulic 
cylinders,  and  on  the  other  the  nature  of  the  metal 
used,  cast-iron,  is  not  such  as  to  warrant  the  assump- 
tion of  flexibility,  as  in  a  thin  shell.  Most  of  the  for- 
mulas used  for  this  class  of  cylinder  are  empirical  and 
founded  on  modern  practice. 

Van  Bur  en's  formula  for  steam  cylinders  is  : 

*  t=.0001pd+.l5\/d (21) 

*  See  Whitham's  "Steam  Engine  Design,"  p.  27. 


STEAM  CYLINDERS. 


43 


A  formula  which  the  writer  has  developed  is  some- 
what similar  to  Van  Buren's. 

Let  s'  =  tangential  stress  due  to  internal  pressure. 
Then  by  equation  for  thin  shells 


Let  s"  be  an  additional  tensile  stress  due  to  distor- 
tion of  the  circular  section  at  any  weak  point. 

Then  if  we  regard  one-half  of  the  circular  section 
as  a  beam  fixed  at  A  and  B  (Fig.  11)  and  assume  the 
maximum  bending  moment  as  at  C  some  weak  point, 
the  tensile  stress  on  the  outer 
fibres  at  C  due  to  the  bending 

will  be  proportional   to  =2* 

i 

by  the  laws  of  flexure,  or 


where  c  is  some    unknown 
constant. 

The  total  tensile  stress  at 
C  will  then  be 


t2 


sr  j_ 

pd*    2d 


.(a) 


*&+&, (22) 


Solving  for  c 
Solving  for  t 


a  form  which  reduces  to  that  of  equation  (13)  when 
c=0. 
An  examination  of    several    engine    cylinders    of 


44  MACHINE  DESIGN. 

standard  manufacture  shows  values  of  c  ranging  from 
.03  to  .10,  with  an  average  value  : 

e=.06. 

The  formula  proposed  by  Professor  Barr,  in  his 
paper  on*  "  Current  Practice  in  Engine  Proportions," 
as  representing  the  average  practice  among  builders  of 
low  speed  engines  is  : 

£=.05  d+.3  inch (23) 

In  Kent's  Mechanical  Engineer's  Pocket  Book,  the 
following  formula  is  given  as  representing  closely 
existing  practice  : 

t=.  0004  dp  +  0.3  inch (24) 

This  corresponds  to  Barr's  formula  if  we  take  p= 125 
pounds  per  square  inch. 

Experiments  f  made  at  the  Case  School  of  Applied 
Science  in  1896-97  throw  some  light  on  this  subject. 
Cast  iron  cylinders  similar  to  those  used  on  engines 
were  tested  to  failure  by  water  pressure.  The  cylinders 
varied  in  diameter  from  six  to  twelve  inches  and  in 
thickness  from  one-half  to  three-quarters  inches. 

Contrary  to  expectations  most  of  the  cylinders  failed 
by  tearing  around  a  circumference  just  inside  the 
flange.  (See  Fig.  12). 


*  Transactions  A.  S.  M.  E.,  vol.  xviii,  p.  741. 
t  Transactions  A.  S.  M.  E.  vol.  xix. 


STEAM  CYLINDERS. 
TABLE  XI. 


45 


No. 

Diam. 
d 

Pres- 
sure. 
P 

Thick- 
ness. 
t 

Line 
of 
Failure. 

Formulas  Used. 

Strength 
of 
Test-bar. 

18 

s=^ 

14 

s=% 

a 
c= 

a 

12.16 

800 

.70 

Circum. 

6940 

3470 

.046 

18000  Ibs. 

d 

12.45 

700 

.56 

Longi. 

7780 



.047 

24000  Ibs. 

e 

9.12 

1325 

.61 

Circum. 

9900 

4950 

.048 

24000  Ibs. 

f 

6.12 

2500 

.65 

Circum. 

11800 

5900 

.055 

24000  Ibs. 

1 

9.58 

600 

.402 

Longi. 

7150 



.049 

24000  Ibs. 

2 

9.375 

1050 

.573 

Circum. 

8590 

4300 

.055 

24000  Ibs. 

3 

9.13 

975 

.596 

Circum. 

7470 

3740 

.072 

24000  Ibs. 

4 

12.53 

700 

.571 

Longi. 

7680 



.048 

24000  Ibs. 

5 

12.56 

875 

.531 

Circum. 

10350 

5180 

.028 

24000  Ibs. 

Average  of  c=.05 

Table  XI  gives  a  summary  of  the  results. 

Out  of  nine  cylinders  so  tested,  only  three  failed  by 
splitting  longitudinally. 

This  appears  to  be  due  to  two  causes.  In  the  first 
place,  the  flanges  caused  a  bending  moment  at  the 
junction  with  the  shell  due  to  the  pull  of  the  bolts. 
In  the  second  place,  the  fact  that  the  flanges  were 
thicker  than  the  shell  caused  a  zone  of  weakness  near 
the  flange  due  to  shrinkage  in  cooling,  and  the  pres- 
ence of  what  founders  call  "  a  hot  spot." 

The  stresses  figured  from  formula  (14)  in  the  cases 
where  the  failure  was  on  a  circumference,  are  from 
one-fifth  to  one-sixth  the  tensile  strength  of  the  test 
bar. 


4:6  MACHINE  DESIGN. 

The  strength  of  a  chain  is  the  strength  of  the 
weakest  link,  and  when  the  tensile  stress  exceeded  the 
strength  of  the  metal  near  some  blow  hole  or  "hot 
spot,"  tearing  began  there  and  gradually  extended 
around  the  circumference. 

Values  of  c  as  given  by  equation  (a)  have  been  cal- 
culated for  each  cylinder,  and  agree  fairly  well,  the 
average  value  being  c=.05. 

To  the  criticism  that  most  of  the  cylinders  did  not 
fail  by  splitting,  and  that  therefore  formulas  (a)  and 
(22)  are  not  applicable,  the  answer  would  be  that  the 
chances  of  failure  in  the  two  directions  seem  about 
equal,  and  consequently  we  may  regard  each  cylinder 
as  about  to  fail  by  splitting  under  the  final  pressure. 

If  we  substitute  the  average  value  of  c=.05  and  a 
safe  value  of  s=2000,  formula  (21)  reduces  to  : 


d       d     I          p* 


*  Subsequent  experiments  made  at  the  Case  School 
in  1904  show  the  effect  of  stiffening  the  flanges  by 
brackets. 

Four  cylinders  were  tested,  each  being  10  inches 
internal  diameter  by  20  inches  long  and  having  a 
thickness  of  about  f  inches.  The  flanges  were  of  the 
same  thickness  as  the  shell  and  were  re-enforced  by 
sixteen  triangular  brackets  as  shown  in  Fig.  13. 

The  fractures  were  all  longitudinal  there  being  but 
little  of  the  tearing  around  the  shell  which  was  so 
marked  a  feature  of  the  former  experiments.  This 
shows  that  the  brackets  served  their  purpose. 

Table  XII  gives  the  results  of  the  tests  and  the 
calculated  values  of  c. 

*  Machinery,  N.  Y.,  Nov.  1905. 


FIG.  12.     FRACTURED  CYLINDER. 


FIG.  13.     FRACTURED  CYLINDER. 


STEAM  CYLINDERS. 

TABLE  XII. 

BURSTING  PRESSURE  OF  CAST-IRON  CYLINDERS. 


Internal 

Average 

Bursting 

Value 

pd 

Diameter. 

Thickness. 

Pressure. 

of  c. 

9f 

10.125 

0.766 

1350 

.0213 

9040 

10.125 

0.740 

1400 

.0152 

10200 

10.125 

0.721 

1350 

.0126 

9735 

10.125 

0.720 

1200 

.0177 

9080 

Average  value  of  c=.0167. 

Comparing  the  values  in  the  above  table  with  those 
in  Table  XI  we  find  c  to  be  only  one-  third  as  large. 

The  tensile  strength  of  the  metal  in  the  last  four 
cylinders,  as  determined  from  test  bars,  was  only 
14000  Ibs.  per  sq.  in. 

Comparison  with  the  values  of  £due  to  direct  tension 
as  given  by  the  formula 


shows  that  in  a  cylinder  of  this  type  about  one-third 
of  the  stress  is  "  accidental"  and  due  to  lack  of  uni- 
formity in  the  conditions.  In.  Table  XI  about  two- 
thirds  must  be  thus  accounted  for. 

PROBLEMS. 

1.  Referring  to  Table  XI,  verify  in  at  least  three  experiments 
the  values  of  8  and  c  as  there  given.     Do  the  same  in  Table 
XII. 

2.  The  steam  cylinder  of  a  Baldwin  locomotive  is  22  ins.  in 
diameter  and  1.25  ins,  thick,    Assuming  125  Ibs.  gauge  pres- 


48  MACHINE  DESIGN. 

sure,  find  the  value  of  c.     Calculate  thickness  by  Van  Buren's 
and  Barr's  formulas. 

3.  Determine  proper  thickness  for  cylinder  of  cast-iron,  if 
the  diameter  is  38  inches  and  the  steam  pressure  100  Ibs.  by 
formulas  13,  21,  23,  24  and  25. 

4.  The  cylinder  of  a  stationary  engine  has  internal  diameter 
=12  in.  and  thickness  of  shell  =1  in.     Find  the  value  of  c  for 
jp=120  Ibs.  per  sq.  in. 

17.  Thickness  of  Flat  Plates.  An  approximate 
formula  for  the  thickness  of  flat  cast-iron  plates  may 
be  derived  as  follows  : 

Let  Z= length  of  plate  in  inches. 

b= breadth  of  plate  in  inches. 
t= thickness  of  plate  in  inches. 
p= intensity  of  pressure  in  pounds. 
S  =  modulus  of  rupture  Ibs.  per  sq.  in. 

A  plate  which  is  supported  or  fastened  at  all  four 
edges  is  constrained  so  as  to  bend  in  two  directions  at 
right  angles.  Now  if  we  suppose  the  plate  to  be 
represented  by  a  piece  of  basket  work  with  strips 
crossing  each  other  at  right  angles  we  may  consider 
one  set  of  strips  as  resisting  one  species  of  bending 
and  the  other  set  as  resisting  the  other  bending.  We 
may  also  consider  each  set  of  strips  as  carrying  a 
fraction  of  the  total  load.  The  equation  of  condition 
is  that  each  pair  of  strips  must  have  a  common  deflec- 
tion at  the  crossing. 

Suppose  the  plate  to  be  divided  lengthwise  into  flat 
strips  an  inch  wide  I  inches  long,  and  suppose  that  a 
fraction  p'  of  the  whole  pressure  causes  the  bending  of 
these  strips. 


THICKNESS  OF  FLAT  PLATES.  49 

Regarding  the  strips  as  beams  with  fixed  ends  and 
uniformly  loaded  : 


bh* 
and  the  thickness  necessary  to  resist  bending  is  : 


In  a  similar  manner,  if  we  suppose  the  plate  to  be 
divided  into  transverse  strips  an  inch  wide  and  b  inches 
long,  and  suppose  the  remainder  of  the  pressure^—  p' 
equals  p"  to  cause  the  bending  in  this  direction,  we 
shall  have  : 


t  —  I 


But  as  all  these  strips  form  one  and  the  same  plate 
the  ratio  of  p'  to  p"  must  be  such  that  the  deflection 
at  the  center  of  the  plate  may  be  the  same  on  either 
supposition.  The  general  formula  for  deflection  in  this 
case  is 

WT 
ZSIEI 

and  I=Tn  for  each  set  of  strips.     Therefore  the  deflec- 
tion is  proportional  to  ^-  and  ^-jr-  m  the  two  cases. 

But  p'  +  p"  =  p 

Solving  in  these  equations  for  pf  and  p" 

,_  pb* 

V 


50  MACHINE  DESIGN. 

Substituting  these  values  in  (a)  and  (b)  : 


t=!»., P. 


As  l>b  usually,  equation  (27)  is  the  one  to  be  used. 
If  the  plate  is  square  l=b  and 


.(88) 


If  the  plate  is  merely  supported  at  the  edges  then 
formulas  (26)  and  (2Y)  become  : 
For  rectangular  plate  : 


/9QN 
ZM%- 

For  square  plate  : 


A  round  plate  may  be  treated  as  square,  with 
side  =  diameter,  without  sensible  error. 

The  preceding  formulas  can  only  be  regarded  as 
approximate.  G-rashof  has  investigated  this  subject 
and  developed  rational  formulas  but  his  work  is  too 
long  and  complicated  for  introduction  here.  His  for- 
mulas for  round  plates  are  as  follows  : 

Round  plates  : 
Supported  at  edges  : 


Fixed  at  edges  : 

'^1.  .     .(32) 


THICKNESS  OF  FLAT  PLATES.  51 

where  t  and  p  are  the  same  as  before,  d  is  the  diameter 
in  inches  and  S  is  the  safe  tensile  strength  of  the 
material. 

Comparing  these  formulas  with  (28)  and  (30)  for 
square  plates,  they  are  seen  to  be  nearly  identical  if 
allowance  is  made  for  the  difference  in  the  value  of  S. 

Experiments  made  at  the  Case  School  of  Applied 
Science  in  1896-97  on  rectangular  cast  iron  plates  with 
load  concentrated  at  the  center  gave  results  as  follows  : 
Twelve  rectangular  plates  planed  on  one  side  and  each 
having  an  unsupported  area  of  10  by  15  inches  were 
broken  by  the  application  of  a  circular  steel  plunger 
one  inch  in  diameter  at  the  geometrical  center  of  each 
plate.  The  plates  varied  in  thickness  from  one-half 
inch  to  one  and  one-eighth  inches.  Numbers  1  to  6 
were  merely  supported  at  the  edges,  while  the  remain- 
ing six  were  clamped  rigidly  at  regular  intervals 
around  the  edge. 

To  determine  the  value  of  S,  the  modulus  of  rup- 
ture of  the  material,  pieces  were  cut  from  the  edge  of 
the  plates  and  tested  by  cross-breaking.  The  average 
value  of  S  from  seven  experiments  was  found  to  be 
33000  Ibs.  per  sq.  in. 

In  Table  XIII  are  given  the  values  obtained  for  the 
breaking  load  W  under  the  different  conditions. 

If  we  assume  an  empirical  formula  : 


and  substitute  given  values  of  S,   I  and   b  we  have 
nearly  : 

W  =  100kt2  .........  (b) 

Substituting  values  of  W  and  t  from  the  Table  XIII 
we  have  the  values  of  k  as  given  in  the  last  column, 


52 


MACHINE  DESIGN. 


If  we  average  the  values  for  the  two  classes  of  plates 
and  substitute  in  (a)  we  get  the  following  empirical 
formulas  : 

For  breaking  load  on  plates  supported  at  the  edges 
and  loaded  at  the  center  : 

(31) 


and  for  similar  plates  with  edges  fixed  : 

TT=442^ (32) 

£  in  both  formulas  is  the  modulus  of  rupture. 

TABLE  XIII. 

CAST  IRON  PLATES  10x15  INS. 


Thickness 

Breaking 

Constant. 

No. 

Load. 

t 

W 

k 

1 

.562 

7500 

237 

2 

.641 

11840 

288 

3 

.745 

14800 

267 

4 

.828 

21900 

320 

5 

1.040 

31200 

289 

6 

1.120 

31800 

254 

7 

.481 

9800 

424 

8 

.646 

17650 

422 

9 

.769 

26400 

446 

10 

.881 

33400 

430 

11 

1.020 

47200 

454 

12 

1.123 

59600 

477 

Those  plates  which  were  merely  supported  at  the 
edges  broke  in  three  or  four  straight  lines  radiating 
from  the  center.  Those  fixed  at  the  edges  broke  in 
four  or  five  radial  lines  meeting  an  irregular  oval 
inscribed  in  the  rectangle.  Number  12  however  failed 
by  shearing,  the  circular  plunger  making  a  circular 
hole  in  the  plate  with  several  radial  cracks. 


THICKNESS  OF  FLAT  PLATES.  53 

Some  tests  were  made  in  the  spring  of  1906  at  the 
Case  School  laboratories  by  Messrs.  Hill  and  Nadig  on 
the  strength  of  flat  cast-iron  plates  under  uniform 
hydraulic  pressure. 

The  plates  tested  were  of  soft  gray  iron,  having  a 
low  tensile  strength  of  about  12000   Ibs.  per  square 
inch,  and  were  of  the  following  sizes  : 
12  by  12  by  f  inches. 
12  by  12  by  1  inches. 
12  by  18  by  1.25  inches. 
12  by  18  by  |f  inches. 
These  burst  at  the  following  pressures  respectively  : 

375  Ibs.  675  Ibs.  650  Ibs.  450  Ibs. 
The  fractures  started  at  the  center  of  the  plates  and 
ran  to  the  sides  in  irregular  lines.  The  square  plates 
were  somewhat  weaker  than  would  have  been  expected 
from  the  formula  and  the  rectangular  plates  somewhat 
stronger. 

PROBLEMS. 

1.  Calculate  the  thickness  of  a  steam-chest  cover  8  X  12 
inches  to  sustain  a  pressure  of  90  Ibs.  per  sq.  inch  with  a  factor 
of  safety  =10. 

2.  Calculate  the  thickness  of  a  circular  manhole  cover  of 
cast-iron  18  inches  in  diameter  to  sustain  a  pressure  of  150  Ibs. 
per  sq.  inch  with  a  factor  of  safety =8,  regarding  the  edges  as 
merely  supported. 

3.  Determine  the  probable  breaking  load  for  a  plate  18  by 
24  in.   loaded  at  the  center,  (a)  when  edges  are  fixed,     (b) 
When  edges  are  supported. 

4.  In  experiments  on  steam  cylinders,  a  head  12  inches  in 
diameter  and  1.18  inches  thick  failed  under  a  pressure  of  900 
Ibs.  per  sq.  in.     Determine  the  value  of  S  by  formula  (28). 


CHAPTER  IV. 


FASTENINGS. 

18.  Bolts  and  Nuts.  Tables  of  dimensions  for  U.  S. 
standard  bolt  heads  and  nuts  are  to  be  found  in  most 
engineering  hand-books  and  will  not  be  repeated  here. 

These  proportions  have  not  been  generally  adopted 
on  account  of  the  odd  sizes  of  bar  required.  The 
standard  screw-thread  has  been  quite  generally  ac- 
cepted as  superior  to  the  old  V- thread. 

Roughly  the  diameter  at  root  of  thread  is  0. 83  of  the 
outer  diameter  in  this  system,  and  the  pitch  in  inches 
is  given  by  the  formula 

p  =  .24i/d+.625-.lT5.     ....     .(33) 

where  d=  outer  diameter. 

TABLE  XIV. 

SAFE  WORKING  STRENGTH  OF  IRON  OR  STEEL  BOLTS. 


Diam. 
of 
Bolt. 

Inch. 

Thr'ds 
per 
Inch. 

No. 

Diam. 
at 
Root  of 
Thread. 
Inches. 

Area 
at 
Root  of 
Thread. 
Sq.  In. 

Safe  Load  in 
Tension.    Lb. 

Safe  Load  in 
Shear.    Lb. 

5000    Ib. 
per  sq.  in. 

7500    Ib. 
per  sq.  in. 

4000    Ib. 
per  sq.  in. 

6000    Ib. 
per  sq.  in 

* 

20 

.185 

.0269 

135 

202 

196 

294 

A 

18 

.240 

.0452 

226 

340 

306 

460 

I 

16 

.294 

.0679 

340 

510 

440 

660 

T7* 

14 

.344 

.0930 

465 

695 

600 

900 

i 

13 

.400 

.1257 

628 

940 

785 

1175 

BOLTS  AND  NUTS. 


55 


TABLE  XIV  (Continued). 
SAFE  WORKING  STRENGTH  OF  IRON  OR  STEEL  BOLTS. 


Diam. 
of 
Bolt. 

Inch. 

Thr'ds 
per 
Inch. 

No. 

Diam. 
at 
Root  of 
Thread. 
Inches. 

Area 
at 
Root  of 
Thread. 
Sq.  In. 

Safe  Load  in 
Tension.    Lb. 

Safe  Load  in 
Shear.    Lt. 

5000  Ib. 
per  sq.  in. 

7500  Ib. 
per  sq.  in. 

4000  Ib, 
per  sq.  in. 

6000  Ib. 
per  sq.  in. 

& 

12 

.454 

.162 

810 

1210 

990 

1485 

i 

11 

.507 

.202 

1010 

1510 

1230 

1845 

i 

10 

.620 

.302 

1510 

2260 

1770 

2650 

t 

9 

.731 

.420 

2100 

3150 

2400 

3600 

i 

8 

.837 

.550 

2750 

4120 

3140 

4700 

H 

7 

.940 

.694 

3470 

5200 

3990 

6000 

1* 

7 

1.065 

.891 

4450 

6680 

4910 

7360 

if 

6 

1.160 

1.057 

5280 

7920 

5920 

7880 

H 

6 

1.284 

1.295 

6475 

9710 

7070 

10600 

tt 

51 

1.389 

1.515 

7575 

11350 

8250 

12375 

If 

5 

1.490 

1.744 

8720 

13100 

9630 

14400 

U 

5 

1.615 

2.049 

10250 

15400 

11000 

16500 

2 

4| 

1.712 

2.302 

11510 

17250 

12550 

18800 

The  shearing  load  is  calculated  from  the  area  of  the 
body  of  the  bolt. 

Bolts  may  be  divided  into  three  classes  which  are 
given  in  the  order  of  their  merit. 

1.  Through  bolts,  having  a  head  on  one  end  and  a 
nut  on  the  other. 

2.  Stud  bolts,  having  a  nut  on  one  end  and  the  other 
screwed  into  the  casting. 

3.  Tap  bolts  or  screws  having  a  head  at  one  end  and 
the  other  screwed  into  the  casting. 


56  MACHINE  DESIGN. 

The  principal  objection  to  the  last  two  forms  and 
especially  to  (3)  is  the  liability  of  sticking  or  breaking 
off  in  the  casting. 

Any  irregularity  in  the  bearing  surfaces  of  head 
or  nut  where  they  come  in  contact  with  the  casting, 
causes  a  bending  action  and  consequent  danger  of 
rupture. 

This  is  best  avoided  by  having  a  slight  annular 
projection  on  the  casting  concentric  with  the  bolt  hole 
and  finishing  the  flat  surface  by  planing  or  counter- 
boring. 

Counter-boring  without  the  projection  is  a  rather 
slovenly  way  of  overcoming  the  difficulty. 

When  bolts  or  studs  are  subjected  to  severe  stress 
and  vibration,  it  is  well  to  turn  down  the  body  of  the 
bolt  to  the  diameter  at  root  of  thread,  as  the  whole 
bolt  will  then  stretch  slightly  under  the  load. 

A  check  nut  is  a  thin  nut  screwed  firmly  against  the 
main  nut  to  prevent  its  working  loose,  and  is  commonly 
placed  outside. 

As  the  whole  load  is  liable  to  come  on  the  outer  nut, 
it  would  be  more  correct  to  put  the  main  nut  outside. 
(Prove  this  by  figure.) 

After  both  nuts  are  firmly  screwed  down,  the  outer 
one  should  be  held  stationary  and  the  inner  one  reversed 
against  it  with  what  force  is  deemed  safe,  that  the 
greater  reaction  may  be  between  the  nuts. 

Numerous  devices  have  been  invented  for  the  purpose 
of  holding  nuts  from  working  loose  under  vibration 
but  none  of  them  are  entirely  satisfactory. 

Probably  the  best  method  for  large  nuts  is  to  drive 
a  pin  or  cotter  entirely  through  nut  and  bolt. 

A  flat  plate,  cut  out  to  embrace  the  nut  and  fastened 
to  the  casting  by  a  machine  screw,  is  often  used. 


MACHINE  SCREWS. 


57 


19.  Machine  Screws.  A  screw  is  distinguished  from 
a  bolt  by  having  a  slotted,  round  head  instead  of  a 
square  or  hexagon  head. 

The  head  may  have  any  one  of  four  shapes,  the 
round,  fillister,  oval  fillister  and 
flat  as  shown  in  Fig.  14.  A 
committee  of  the  American 
Society  of  Mechanical  Engineers 
has  recently  recommended  cer- 
tain standards  for  machine 
screws.*  The  form  of  thread 
recommended  is  the  U.  S.  Stand- 
ard or  Sellers  type  with  provision 
for  clearance  at  top  and  bottom 
to  insure  bearing  on  the  body  of 
the  thread. 

The  sizes  and  pitches  recommended  are  as  follows  : 

TABLE  XV. 
MACHINE  SCREWS. 


Standard  Diameter. 

.070 

.085 

.100 

.110 

.125 

.140 

.165 

.190 

.215 

28 

.240 

.250 

.270 

.320 

.375 

Threads  per  inch. 

72 

64 

56 

48 

44 

40 

36 

32 

24 

24 

22 

20 

16 

Reference  is  made  to  the  report  itself  for  further 
details  of  heads,  taps,  etc. 

20.  Eye  Bolts  and  Hooks.  In  designing  eye  bolts 
it  is  customary  to  make  the  combined  sectional  area  of 
the  sides  of  the  eye  one  and  one  half-times  that  of  the 
bolt  to  allow  for  obliquity  and  an  uneven  distribution 
of  stress. 

Large  hooks  should  be  designed  to  resist  combined 


*  Trans.  A.  S.  M.  E.,  Vol.  xxvii. 


58  MACHINE  DESIGN. 

bending  and  tension  ;  the  bending  moment  is  equal  to 
the  load  multiplied  by  the  longest  perpendicular  from 
the  center  line  of  hook  to  line  of  load. 

The  tension  due  to  this  bending  must  be  added  to  the 
direct  tension  and  the  body  of  the  hook  designed  ac- 
cordingly. 

PROBLEMS. 

1.  Discuss  the  effect  of  the  initial  tension  caused  by  the 
screwing  up  of  the  nut  on  the  bolt,  in  the  case  of  steam  fittings, 
etc.  ;  i.  e.  should  this  tension  be  added  to  the  tension  due  to 
the  steam  pressure,  in  determining  the  proper  size  of  bolt  ? 

2.  Determine  the  number  of  f  inch  steel  bolts  necessary  to 
hold  on  the  head  of  a  steam  cylinder  15  inches  diameter,  with 
the  internal  pressure  90  pounds  per  square  inch,  and  factor  of 
safety =12. 

3.  Show  what  is  the  proper  angle  between  the  handle  and 
the  jaws  of  a  fork  wrench. 

(1)  If  used  for  square  nuts  ; 

(2)  If  used  for  hexagon  nuts  ;  illustrate  by  figure. 

4.  Determine  the  length  of  nut  theoretically  necessary  to 
prevent  stripping  of  the  thread,  in  terms  of  the  outer  diameter 
of  the  bolt. 

(1)  With  U.  S.  standard  thread.  . 

(2)  With  square  thread  of  same  depth. 

5.  Design  a  hook  with  a  swivel  and  eye  at  the  top  to  hold  a 
load  of  one  ton  with  a  factor  of  safety  5,  the  center  line  of 
hook  being  three  inches  from  line  of  load,  and  the  material 
wrought  iron. 

21.  Riveted  Joints.  Eiveted  joints  may  be  divided 
into  two  general  classes  :  lap  joints  where  the  two 
plates  lap  over  each  other,  and  butt  joints  where  the 
edges  of  the  plates  butt  together  and  are  joined  by 
over-lapping  straps  or  welts.  If  the  strap  is  on  one 


RIVETED  JOINTS. 


59 


D 


c 


side  only,  the  joint  is  known  as  a  butt  joint  with  one 
strap  ;  if  straps  are  used  inside  and  out  the  joint  is 
called  a  butt  joint  with  two  straps.  Butt  joints  are 
generally  used  when  the  material  is  more  than  one 
half  inch  thick. 

Any  joint  may  have  one,  two  or  more  rows  of  rivets 
and  hence  be  known  as  a  single  riveted  joint,  a  double 
riveted  joint,  etc. 

Any  riveted  joint  is 
weaker  than  the  origi- 
nal plate,  simply 

CP^^        A   ~-     C\~  O  -/Z?  oecause   holes   cannot 
^^^  be  punched  or  drilled 

C-^'N  /^v      s~\      r     in  the  plate   for    the 

^sL/  i  i     introduction  of  rivets 

without     removing 
some  of  the  metal. 

Fig.      15     shows    a 
double  riveted  lap  joint  and  Fig.  16  a  single  riveted 
butt  joint  with  two  straps. 
Eiveted  joints  may  fail  in  any  one  of  four  ways  : 

1.  By  tearing  of  the 
plate  along  a  line  of 
rivet  holes,  as  at  AB, 
Fig.  15. 

2.  By    shearing 
the  rivets. 

3.  By  crushing   and  [ 
wrinkling  of  the  plate 
in  front  of  each  rivet 
as  at  C,  Fig.  15,  thus 
causing  leakage. 

4.  By  splitting  of  the  plate  opposite  each  rivet  as 
at  D,  Fig.  15.     The  last  manner  of  failure  may  be  pre- 


15. 


of 


— 


o 

O 

=1 
o 

0 

O 

0 

==1 

Fig.  16. 

60 


MACHINE  DESIGN. 


vented  by  having  a  sufficient  distance  from  the  rivet 
to  the  edge  of  the  plate.  Practice  has  shown  that  this 
distance  should  be  at  least  equal  to  the  diameter  of  a 
rivet. 

Experience  has  shown  that  lap  joints  in  plates  of 
even  moderate  thickness  are  dangerous  on  account  of 
the  liability  of  hidden  cracks.  Several  disastrous 
boiler  explosions  have  resulted  from  the  presence  of 
cracks  inside  the  joint  which  could  not  be  detected  by 
inspection.  The  fact  that  one  or  both  plates  are  out 
of  the  line  of  pull  brings  a  bending  moment  on  both 
plates  and  rivets. 

Some  boiler  inspectors  have  gone  so  far  as  to  condemn 
lap-joints  altogether. 
Let        £= thickness  of  plate. 

d= diameter  of  rivet-hole. 

p= pitch  of  rivets. 

n= number  of  rows  of  rivets. 

T= tensile  strength  of  plate. 

C  =  crushing  strength  of  plate  or  rivet. 

S= Shearing  strength  of  rivet. 
Average  values  of  the  constants  are  as  follows  : 


Material. 

T 

C 

8 

Wrought  Iron  

50  000 

80  000 

40  000 

Soft  Steel.., 

56  000 

90  000 

45  000 

The  values  of  the  constants  given  above  are  only 
average  values  and  are  liable  to  be  modified  by  the 
exact  grade  of  material  used  and  the  manner  in  which 
it  is  used. 


LAP  JOINTS.  61 

The  tensile  strength  of  soft  steel  is  reduced  by 
punching  from  three  to  twelve  per  cent  according  to 
the  kind  of  punch  used  and  the  width  of  pitch.  The 
shearing  strength  of  the  rivets  is  diminished  by  their 
tendency  to  tip  over  or  bend  if  they  do  not  fill  the 
holes,  while  the  bearing  or  compression  is  doubtless 
relieved  to  some  extent  by  the  friction  of  the  joint. 
The  values  given  allow  roughly  for  these  modifica- 
tions. 

22.  Lap  Joints.  This  division  also  includes  butt 
joints  which  have  but  one  strap. 

Let  us  consider  the  shell  as  divided  into  strips  at 
right  angles  to  the  seam  and  each  of  a  width  =p. 
Then  the  forces  acting  on  each  strip  are  the  same  and 
we  need  to  consider  but  one  strip. 

The  resistance  to  tearing  across  of  the  strip  between 
rivet  holes  is  (p—d)tT.     ......  (a) 

and  this  is  independent  of  the  number  of  rows  of 
rivets. 

The  resistance  to  compression  in  front  of  rivets  is 

ndtC  ..........  (b) 

and  the  resistance  to  shearing  of  the  rivets  is 


(c) 


If  we  call  the  tensile  strength  T=  unity  then  the 
relative  values  of  C  and  $  are  1.6  and  0.8  respectively. 

Substituting  these  relative  values  of  T,  C  and  S 
in  our  equations,  by  equating  (b)  and  (c)  and  reducing 
we  have  d=2.55t  ........  (34) 

Equating  (a)  and  (c)  and  reducing  we  have 

(35) 


02  MACHINE  DESIGN. 

Or  by  equating  (a)  and  (b) 

p  =  d+1.6nd  .......  (36) 

These  proportions  will  give  a  joint  of  equal  strength 
throughout,  for  the  values  of  constants  assumed. 

23.  Butt  Joints  with  two  Straps.  In  this  case  the 
resistance  to  shearing  is  increased  by  the  fact  that  the 
rivets  must  be  sheared  at  both  ends  before  the  joint 
can  give  away.  Experiment  has  shown  this  increase 
of  shearing  strength  to  be  about  85  per  cent  and  we 
can  therefore  take  the  relative  value  of  S  as  1.5  for 
butt  joints. 

This  gives  the  following  values  for  d  and  p 

dUl.36*  ........  (37) 

p=d+l.l£f  ......  (38) 

p=d+l.Qnd  .......  (39) 

In  the  preceding  formulas  the  diameter  of  hole  and 
rivet  have  been  assumed  to  be  the  same. 

The  diameter  of  the  cold  rivet  before  insertion  will 
be  TV  inches  less  than  the  diameter  given  by  the 
formulas. 

Experiments  made  in  England  by  Prof.  Kennedy 
give  the  following  as  the  proportions  of  maximum 
strength  : 

Lap  joints  cZ=2.33£ 


Butt  joints  d=l.St 


24.  Efficiency  of  Joints.  The  efficiency  of  joints 
designed  like  the  preceding  is  simply  the  ratio  of  the 
section  of  plate  left  between  the  rivets  to  the  section 


BETT  JOINTS  WITH  UNEQUAL  STRAPS. 


63 


of  solid  plate,  or  the  ratio  of  the  clear  distance  between 
two  adjacent  rivet  holes  to  the  pitch.  From  formula 
(35)  we  thus  have. 

........  (40) 


This  gives  the  efficiency  of  single,  double  and  triple 
riveted  seams  as 

.615,  .762  and  .828  respectively. 

Notice  that  the  advantage  of  a  double  or  triple 
riveted  seam  over  the  single  is  in  the  fact  that  the  pitch 
bears  a  greater  ratio  to  the  diameter  of  a  rivet,  and 
therefore  the  proportion  of  metal  removed  is  less. 

25.  Butt  Joints  with  unequal  Straps.  One  joint  in 
common  use  requires  special  treatment. 

It  is  a  double-riveted  butt  joint  in  which  the  inner 
strap  is  made  wider 
than   the   outer    and 
an  extra  row  of  rivets 
added,  whose  pitch  is 
double    that    of    the 
original  seam  ;  this  is 
sometimes 
diamond 
See  Fig.  17. 

This  outer 


row  of 
rivets  is  then  exposed 
to  single  shear  and 
the  original  rows  to 
double  shear. 

Consider  a  strip  of 
plate  of  a  width  =  2p. 


c  a  1 led 
riveting. 


i_ 


Fig.  17. 


Then  the  resistance  to  tearing  along  the  outer  row  of 
rivets  is  (2p-d)tT 


64  MACHINE  DESIGN. 

As  there  are  five  rivets  to  compress  in  this  strip  the 
bearing  resistance  is 


As  there  is  one  rivet  in  single  shear  and  four  in 
double  shear  the  resistance  to  shearing  is 


+  (4  X  1.85)       d*S  =  6. 

Solving  these  equations  as  in  previous  cases,  we 
have  for  this  particular  joint 

d=l.&2t       .......  (41) 


(42) 
(43) 


26.  Practical  Rules.  The  formulas  given  above 
show  the  proportions  of  the  usual  forms  of  joints  for 
uniform  strength. 

In  practice  certain  modifications  are  made  for 
economic  reasons.  To  avoid  great  variation  in  the 
sizes  of  rivets  the  latter  are  graded  by  sixteenths  of  an 
inch,  making  those  for  the  thicker  plates  considerably 
smaller  than  the  formula  would  allow,  and  the  pitch 
is  then  calculated  to  give  equal  tearing  and  shearing 
strength. 

Table  XVI  shows  what  may  be  considered  average 
practice  in  this  country  for  lap-joints  with  steel  plates 
and  rivets. 


PRACTICAL  RULES. 


65 


TABLE  XVI. 

RIVETED  LAP  JOINTS. 


Thick- 
ness of 
Plate. 

Diam. 
of 
Rivet. 

Diam. 
of 
Hole. 

Pitch. 

Efficiency  of  Plate. 

Single. 

Double. 

Single. 

Double. 

i 

* 

A 

If 

If 

.59 

.68 

T56 

f 

H 

If 

2i 

.58 

.68 

1 

f 

if 

U 

2* 

.57 

.67 

I7. 

It 

1 

2 

2f 

.56 

.68 

1 

1 

H 

2 

21 

.53 

.67 

The  efficiencies  are  calculated  from  the  strength  of 
plate  between  rivet  holes  and  the  efficiencies  of  the 
rivets  rnay  be  even  lower.  Comparing  these  values 
with  the  ones  given  in  Art.  24  we  find  them  low.  This 
is  due  to  the  fact  that  the  pitches  assumed  are  too 
small.  The  only  excuse  for  this  is  the  possibility  of 
getting  a  tighter  joint. 

TABLE  XVII. 

RIVETED  BUTT  JOINTS. 


Thickness 

Diam. 

Diam. 

Pitch. 

of 

of 

of 

Plate. 

Rivet. 

Hole. 

Single. 

Double. 

Triple. 

i 

t 

if 

3f 

4 

5* 

f 

it 

1 

2f 

3f 

5i 

£ 

* 

if 

2f 

3| 

5i 

* 

if 

1 

2f 

8| 

5 

1 

1 

h* 

2f 

3| 

5 

06  MACHINE  DESIGN. 

Table  XVII  has  been  calculated  for  butt  joints  with 
two  straps.  As  in  the  preceding  table  the  values  of 
the  pitch  are  too  small  for  the  best  efficiency.  The 
tables  are  only  intended  to  illustrate  common  practice 
and  not  to  serve  as  standards.  There  is  such  a  diversity 
of  practice  among  manufacturers  that  it  is  advisable 
for  the  designer  to  proportion  each  joint  according  to 
his  own  judgment,  using  the  rules  of  Arts.  22-25  and 
having  regard  to  the  practical  considerations  which 
have  been  mentioned. 

A  committee  of  the  Master  Steam  Boiler  Makers' 
Association  has  made  a  number  of  tests  on  riveted 
joints  and  reported  its  conclusions*  The  specimens 
were  prepared  according  to  generally  accepted  practice, 
but  on  subjecting  them  to  tension  many  of  them  failed 
by  tearing  through  from  hole  to  edge  of  plate.  The 
committee  recommends  making  this  distance  greater, 
so  that  from  the  center  of  hole  to  edge  of  plate  shall  be 
perhaps  2d  instead  of  1.5<i. 

The  committee  further  found  the  shearing  strength 
of  rivets  to  be  in  pounds  per  square  inch  of  section. 


Single  Shear. 

Double  Shear. 

40000 

78000 

Steel  rivets 

49000 

84000 

Compare  these  values  with  those  given  in  Art.  21. 
Also  note  that  the  factor  for  double  shear  is  1.95  for 
iron  rivets  and  only  1.71  for  steel  rivets  as  against  the 
1.85  given  in  Art.  23.  The  committee  found  that 
machine-driven  rivets  were  stronger  in  double  shear 
than  hand-driven  ones. 


PRACTICAL  RULES.  67 

PROBLEMS. 

1.  Calculate  diameter  and  pitch  of    rivets  for  J  in.   and 
i.  in.  plate   and  compare  results  with  those  in  Table  XVI. 
Criticise  latter. 

2.  Show  the  effect  in  Prob.  1  of  using  iron  rivets  in  steel 
plates. 

3.  Criticise  proportions  of  joints  for  i  in.  and  1  in.  plate  in 
Table  XVII.  by  testing  the  efficiency  of  rivets  and  plates. 

4.  A  cylinder  boiler  5x16  ft.  is  to  have  long  seams  double- 
riveted  laps  and  ring  seams  single  riveted  laps.     If  the  inter- 
nal pressure  is  90  Ibs.  gauge  pressure  and  the  material  soft 
steel,  determine  thickness  of  plate  and  proportion  of  joints. 
The  net  factor  of  safety  at  joints  to  be  five. 

5.  A  marine  boiler  is  11  ft.  6  ins.  in  diameter  and  14  ft  long. 
The  long  seams  are  to  be  diamond  riveted  butt  joints  and  the 
ring  seams  ordinary  double  riveted  butt  joints.     The  internal 
pressure  is  to  be  175  Ibs.  gauge  and  the  material  is  to  be  steel 
of  60,000  Ibs.  tensile  strength.     Determine  thickness  of  shell 
and  proportions  of  joints.     Net  factor  of  safety  to  be  5,  as  in 
Prob.  4. 

6.  Design  a  diamond  riveted  joint  such  as  shown  in  Fig  18 
for  a  steel  plate  J  in.  thick.      Outer  cover  plate  is  J  in.  and 
inner  cover  plate  is  T7^-  inches  thick  ;  the  pitch  of  outer  rows  of 
rivets  to  be  twice  that  of  inner  rows.     Determine  efficiency  of 
joints. 

r\  r\ 


W///////////A 


Fig.  18. 

7.  The  single  lap  joint  with  cover  plate,  as  shown  in  Fig.  19, 
is  to  have  pitch  of  outer  rivets  double  that  of  inner  row,    De- 


68 


MACHINE  DESIGN. 


termine  diameter  and  pitch  of  rivets  for  |  inch  plate  and  the 
efficiency  of  joint. 


27.  Riveted  Joints  for  Narrow  Plates.  The  joints 
which  have  been  so  far  described  are  continuous  and 
but  one  strip  of  a  width  equal  to  the  pitch  or  the  least 
common  multiple  of  several  pitches,  has  been  investi- 
gated. 

When  narrow  plates  such  as  are  used  in  structural 
work  are  to  be  joined  by  rivets,  the  joint  is  designed 
as  a  whole.  Diamond  riveting  similar  to  that  shown 
in  Fig.  17  is  generally  used  and  the  joint  may  be  a 
lap,  or  a  butt  with  double  straps.  The  diameter  of 
rivets  may  be  taken  about  1.5  times  the  thickness  of 
plate  [see  equation  (41)  ],  and  enough  rivets  used  so 
that  the  total  shearing  strength  may  equal  the  tensile 
strength  of  the  plate  at  the  point  of  the  diamond, 
where  there  is  one  rivet  hole.  It  may  be  necessary  to 
put  in  more  rivets  of  a  less  diameter  in  order  to  make 
the  figure  symmetrical. 

The  efficiency  of  the  joint  may  be  tested  at  the  dif- 
ferent rows  of  rivets,  allowing  for  tension  of  plate 
and  shear  of  rivets  in  each  case. 

PROBLEMS. 

1.  Design  a  diamond-riveted  lap-joint  for  a  plate  ten  inches 
wide  and  one-half  inch  thick,  and  calculate  least  efficiency  for 
shear  antf  tension. 


JOINT  PINS. 


69 


2.  A  diamond- riveted  butt-joint  with  two  straps  has  rivets 
arranged  as  in  Fig.  20,  the  plate  being  twelve 
inches  wide  and  three-quarter  inches  thick, 
and  the  rivets  being  one  inch  in  diameter.  If 
the  plate  and  rivets  are  of  steel,  find  the 
probable  ultimate  strength  of  the  following 


o 
o  o 


0__0__0 

o  o  o 

o  o 

o 


parts  : 

(a)  The  whole  plate. 

(6)  All  the  rivets  on  one  side  of  the  joint. 

(c)  The  joint    at  the  point  of  the    dia- 
mond. 

(d)  The  joint  at  the  row  of  rivets  next 
the  point. 


Fig.  20. 


28.  Joint  Pins.     A  joint  pin  is  a   bolt  exposed  to 
double  shear.     If  the  pin  is  loose  in  its  bearings  it 
should  be  designed  with  allowance  for   bending,   by 
adding  from  30  to  50  per  cent  to  the  area  of  cross- 
section  needed  to  resist  shearing  alone.     Bending  of  the 
pin  also  tends  to  spread  apart  the  bearings  and  this 
should  be  prevented  by  having  a  head  and  nut  or  cotter 
on  the  pin. 

If  the  pin  is  used  to  connect  a  knuckle  joint  as  in 
boiler  stays,  the  eyes  forming  the  joint  should  have  a 
a  net  area  50  per  cent  in  excess  of  the  body  of  the  stay, 
to  allow  for  bending  and  uneven  tension,  (see  Eye- 
bolts,  Art.  20.) 

Fig.  21  shows  a  pin  and  angle  joint  for  attaching 
the  end  of  a  boiler  stay  to  the  head  of  the  boiler. 

29.  Cotters.     A  cotter  is  a  key  which  passes  diamet- 
rically through  a  hub  and  its  rod  or  shaft,  to  fasten 


MACHINE  DESIGN. 


them  together,  and  is  so  called  to  distinguish  it  from 
shafting  keys  which  lie  parallel  to  axis  of  shaft. 


u 


1=3 


Fig.  21. 


Fig.  22. 


Its  taper  should  not  be  more  than  4  degrees  or 
about  1  in  15,  unless  it  is  secured  by  a  screw  or  check 
nut. 

The  rod  is  sometimes  enlarged  where  it  goes  in  the 
hub,  so  that  the  effective  area  of  cross-section  where 
the  cotter  goes  through  may  be  the  same  as  in  the 
body  of  the  rod.  (See  Fig.  22.) 

Let :          d  =  diameter  of  body  of  rod. 

d,= diameter  of  enlarged  portion. 

t= thickness    of  cotter,  usually = -^ 

b= breadth  of  cotter. 
I— length  of  rod  beyond  cotter. 

Suppose  that  the  applied  force  is  a  pull  on  the  rod- 
causing  tension  on  the  rod  and  shearing  stress  on  the 
cotter. 

The  effective  area  of  cross  section  of  rod  at'  cotter  is 
ird?     d\_,         .d\ 


COTTERS.  71 


and  this  should  equal  the  area  of  cross-section  of  the 
body  of  rod. 


(44) 


Let  P=pull  on  rod. 

S=  shearing  strength  of  material. 
The  area  to  resist  shearing  of  cotter  is 

•"-*£•$ 

2P 

'•••  b=^s 

The  area  to  resist  shearing  of  rod  is 

fttf-f 

andl=M$  .......  ' 

If  the  metal  of  rod  and  cotter  are  the  same 


Great  care  should  be  taken  in  fitting  cotters  that 
they  may  not  bear  on  corners  of  hole  and  thus  tear  the 
rod  in  two. 

A  cotter  or  pin  subjected  to  alternate  stresses  in  op- 
posite directions  should  have  a  factor  of  safety  double 
that  otherwise  allowed. 

Adjustable  cotters,  used  for  tightening  joints  of 
bearings  are  usually  accompanied  by  a  gib  having  a 


72  MACHINE  DESIGN. 

taper  equal  and  opposite  to  that  of  the  cotter.     (Fig. 

^^  23).     In     designing   these    for 

|  strength   the   two    can   be   re- 

f 1 f — p-j ,        garded   as   resisting   shear  to- 

^    gether. 

/        For  shafting  keys  see  chapter 
\    on  shafting. 

I     I  (          The  split  pin  is  in  the  nature 

I       J  of  a  cotter  but  is   not  usually 

p.     33  expected  to  take  any  shearing 

stress. 

PROBLEMS. 

1.  Design  an  angle  joint  for  a  soft  steel  boiler  stay,  the  pull 
on  stay  being  12000  Ibs.  and  the  factor  of  safety,  six.     Use 
two  standard  angles. 

2.  Determine  the  diameter  of  a  round  cotter  pin  for  equal 
strength  of  rod  and  pin. 

3.  A  rod  of  wrought  iron  has  keyed  to  it  a  piston  18  inches 
in  diameter,  by  a  cotter  of  machinery  steel. 

Required  the  two  diameters  of  rod  and  dimensions  of  cotter 
to  sustain  a  pressure  of  150  pounds  per  square  inch  on  the 
piston.  Factor  of  safety  =  8. 

Design  a  cotter  and  gib  for  connecting  rod  of  engine  men- 
tioned in  Prob.  3,  both  to  be  of  machinery  steel  and  .75  inches 
thick.  (See  Fig.  23.) 


CHAPTER  V. 

SPRINGS. 

30.  Helical  Springs.  The  most  common  form  of 
spring  used  in  machinery  is  the  spiral  or  helical  spring 
made  of  round  brass  or  steel  wire.  Such  springs  may 
be  used  to  resist  extension  or  compression  or  they  may 
be  used  to  resist  a  twisting  moment. 

Tension  and  Compression. 

Let  Li—  length  of  axis  of  spring. 
D=mean  diameter  of  spring. 
I—  developed  length  of  wire. 
d=  diameter  of  wire. 

E=  ratio   ^-. 
d 

n=  number  of  coils. 

P=  tensile  or  compressive  force. 

x=  corresponding  extension  or  compression. 

£=safe  torsional  or  shearing  strength  of  wire. 

=45000  to  60000  for  spring  brass  wire. 

=T5000  to  115000  for  cast  steel  tempered. 
G=  modulus  of  torsional  elasticity. 

—  6000000  for  spring  brass  wire. 

=  12000000  to  15000000  for  cast  steel,  tempered. 
Then  Z  = 


If  the  spring  were  extended  until  the  wire  became 

73 


74  MACHINE  DESIGN. 

straight  it  would  then  be  twisted  n  times,  or  through 
an  angle  =  27m  and  the  stretch  would  be        I  —  L. 
The  angle  of  torsion  for  a  stretch  =x  is  then 


Suppose  that  a  force  P'  acting  at  a  radius  -~    will 

a 

twist  this  same  piece  of  wire  through  an  angle  0  causing 
a  stress  S  at  the  surface  of  the  wire.  Then  will  the 
distortion  of  the  surface  of  the  wire  per  inch  of  length 

be  s=^  and  the  stress  S=^=^£    .     .    .  (b) 

.  r    8    10.2P'DZ  ,  , 

1  -G=s  =  -33*-     ......  (C) 

In  thus  twisting  the  wire  the  force  required  will 
vary  uniformly  from  o  at  the  beginning  to  Pf  at  the 
end  provided  the  elastic  limit  is  not  passed,  and  the 
average  force  will  be 

pf  PfT)B 

=-£-    The  work  done  is  therefore  —  -j  — 

If  the  wire  is  twisted  through  the  same  angle  by 
the  gradual  application  of  the  direct  pressure  P,  com- 
pressing or  extending  the  spring  the  amount  x,  the 
work  done  will  be 

PX 

-3- 


DO 

Substituting  this  value  of  P'  in  (c)  and  solving  for  x  : 
Gd'ffi 

;   «*/  =  -TTr 


10.2  PI 


SQUARE  WIRE.  75 

Substituting  the  value  of  0  from  (a)  and  again  solving 
for  x  : 

10.2P/  (  I— L  )  2  ,, 

If  we  neglect  the  original  obliquity  of  the  wire  then 
l=irDn  and  L=o  and  equation  (e)  reduces  to 

2.55P/D2 

tJO  T~ZZ ~^y — -jj         ••          +  •••••  *  *  (*jbO} 

Making  the  same  approximation  in  equation  (d)  we 
have  P'=P 

i.e. — a  force  P  will  twist  the  wire  through  approxi- 
mately the  same  angle  when  applied  to  extend  or  com- 
press the  spring,  as  if  applied  directly  to  twist  a  piece 
of  straight  wire  of  the  same  material  with  a  lever 

arm=? 

This  may  be  easily  shown  by  a  model* 
The  safe  working  load  may  be  found  by  solving  for 
P'  in  (b)  and  remembering  that  P  =  P' 


~2.55Z>~  2.55.R 

when  S  is  the  safe  shearing  strength. 

Substituting  this  value  of  P  in  (32)  we  have  for  the 
safe  deflection  : 


31.  Square  Wire.     The  value  of    the  stress  for  a 
square  section  is  : 


where  d  is  the  side  of  square. 


76  MACHINE  DESIGN. 

The  distortion   at   the  corners   caused  by  twisting 
through  an  angle  0  is  : 

=  Od 

ll/2 

Equation  (c)  then  becomes : 
SP'Dl 

The  three  principal  equations  (46),  (47)  and  (48)  then 
reduce  to : 


X== — GkP —    •     *    *    •     *     *      (49) 


(51) 

The  square  section  is  not  so  economical  of  material 
as  the  round* 

32.  Experiments.  Tests  made  on  about  1700  tem- 
pered steel  springs  at  the  French  Spring  Works  in 
Pittsburg  were  reported  in  1901  by  Mr.  R.  A.  French.* 
These  were  all  compression  springs  of  round  steel  and 
were  given  a  permanent  set  before  testing  by  being 
closed  coil  to  coil  several  times.  Table  XVIII  gives 
results  of  these  experiments. 

*  Trans.  A.  S.  M.  E.,  Vol.  XXIII. 


•SSOJ1S   3UUB811S 


EXPERIMENTS. 

'888888888888! 


jmtoisaoj, 


>||§||§|||gg|gS88888§8888Sl 
[S  S 1 8 1 8  8  8  S88  8  8  8  8  88  §|  88  8|  8  8  < 


•Snijds 
esoio  ocj 


111 


•,,H—  ,H 


«0  TH  T-KN  <N  <M  1-1  93  i-i  W  OJ  T-I       TH  T-(  r-i  rt  T-I  T-UXN  TI  rt  (M  T-II-I  CO 


m         m  m 


ff  —  ff  = 


1     §3 

CD  lOO 


O»  i    O  O  CD  00  O»  « 


30 


UU3H 


epis^no 


ml 


30 


? «  5»  SI  Si  S  w  S  Sj 


78  MACHINE  DESIGN. 

The  apparent  variation  of  G  in  the  experiments  is 
probably  due  to  differences  in  the  quality  of  steel  and 
to  the  fact  that  the  formula  for  G  in  the  case  of  helical 
springs  is  an  approximate  one. 

The  same  may  be  said  of  the  values  of  S,  but  if  these 
values  are  used  in  designing  similar  springs  one  error 
,will  off-set  the  other. 

In  some  few  cases,  as  in  No.  18,  it  was  necessary  to 
use  an  abnormally  high  value  of  S  to  meet  the  condi- 
tions. This  necessitated  a  special  grade  of  steel,  and 
great  care  in  manufacture.  Such  a  spring  is  not  safe 
when  subjected  to  sudden  and  heavy  loads,  or  to  rapid 
vibrations,  as  it  would  soon  break  under  such  treat- 
ment ;  if  merely  subjected  to  normal  stress,  it  would 
last  for  years. 

Springs  of  a  small  diameter  may  safely  be  subjected 
to  a  higher  stress  than  those  of  a  larger  diameter,  the 
size  of  bar  being  the  same.  The  safe  variation  of  S 
with  R  cannot  yet  be  stated. 

There  is  an  important  limit  which  should  be  here 
mentioned.  Springs  having  too  small  a  diameter  as 
compared  with  size  of  bar  are  subjected  to  so  much 
internal  stress  in  coiling  as  to  weaken  the  steel.  A 
spring,  to  give  good  service,  should  never  have  R  less 
than  3. 

The  size  of  bar  has  much  to  do  with  the  safe  value 
of  S ;  the  probable  explanation  is  this  :  A  large  bar 
has  to  be  heated  to  a  higher  temperature  in  working 
it,  and  in  high  carbon  steel  this  may  cause  deterio- 
ration ;  when  tempered,  the  bath  does  not  affect  it  so 
uniformly,  as  may  be  seen  by  examining  the  fracture 
of  a  large  bar. 

The  above  facts  must  always  be  taken  into  consid- 
eration in  designing  a  spring,  whatever  the  grade  of 


SPRING  IN  TORSION.  79 

steel  used.  A  safe  value  of  S  can  be  determined  only 
by  one  having  an  accurate  knowledge  of  the  physical 
characteristics  of  the  steel,  the  proportions  of  the 
spring,  and  the  conditions  of  use. 

7.  For  a  good  grade  of  steel  the  following  values  of 
S  have  been  found  safe  under  ordinary  conditions  of 
service,  the  value  of  G  being  taken  as  14,500,000. 

VALUES  OF  S. 


*=3 

R=S 

(1  —  s  inch  or  less     

112,000 

85,000 

d  —  jig  inch  to  f  inch  

110,000 

80,000 

(I  —  1|  inch  to  1J  inch     

105,000 

75.000 

For  bars  over  1^  inches  in  diameter  a  stress  of  more 
than  100,000  should  not  be  used.  Where  a  spring  is 
subjected  to  sudden  shocks  a  smaller  value  of  S  is 
necessary. 

As  has  been  noted,  the  springs  referred  to  in  this 
paper  were  all  compression  springs.  Experience  has 
shown  that  in  close  coil  or  extension  springs  the  value 
of  G  is  the  same,  but  that  the  safe  value  of  S  is  only 
about  two-thirds  that  for  a  compression  spring  of  the 
same  dimensions. 

33.  Spring  in  Torsion.  If  a  helical  spring  is  used  to 
resist  torsion  instead  of  tension  or  compression,  the 
wire  itself  is  subjected  to  a  bending  moment.  We  will 
use  the  same  notation  as  in  the  last  article,  only  that 
P  will  be  taken  as  a  force  acting  tangentially  to  the 

circumference  of  the  spring  at  a  distance  -^-  from  the 


80  MACHINE  DESIGN. 

axis,  and  S  will  now  be  the  safe  transverse  strength  of 
the  wire,  having  the  following  values  : 
£  =  60,000  for  spring  brass  wire. 

=  90,000  to  125000  for  cast  steel  tempered. 
.27=9000000  for  spring  brass  wire. 

=  30000000  for  cast  steel  tempered. 
Let  6=  angle  through  which  the  spring  is  turned 
by  P. 

The  bending  moment  on  the  wire  will  be  the  same 

PD 
throughout   and      ''~^~-     This  is  best  illustrated  by  a 

model. 

To  entirely  straighten  the  wire  by  unwinding  the 
spring  would  require  the  same  force  as  to  bend  straight 
wire  to  the  curvature  of  the  helix. 

To  simplify  the  equations  we  will  disregard  the 
obliquity  of  the  helix,  then  will  l=vDn  and  the  radius 

of  curvature  ==!F" 

Let  M=  bending  moment  caused  by  entirely 
straightening  the  wire  ;  then  by  mechanics 

EI=2EI 

and  the  corresponding  angle  through  which  spring  is 
turned  is  2-n-n. 

But  it  is  assumed  that  a  force  P  with  a  radius  —^ 
turns  the  spring  through  an  angle  0. 

.      PD_%EI      0 
2  " "  D      %irn 

EIB   Eie 
~7rDn~   I 


FLAT  SPRINGS.  81 

Solving  for  0 : 

6=PDl  f^ 

and  if  wire  is  round 


The  bending  moment  for  round  wire  will  be 
PD_  Sds 


and  this  will  also  be  the  safe  twisting  moment  that  can 
be  applied  to  the  spring  when  S=  working  strength  of 
wire.  The  safe  angle  of  deflection  is  found  by  sub- 

PD 

stituting  this  value  of  --  in  (52)  : 


Keducing:       0=       .........  (54) 


34.  Flat  Springs.  Ordinary  flat  springs  of  uniform 
rectangular  cross-section  can  be  treated  as  beams  and 
their  strength  and  deflection  calculated  by  the  usual 
formulas. 

In  such  a  spring  the  bending  and  the  stress  are 
greatest  at  some  one  point  and  the  curvature  is  not 
uniform. 

To  correct  this  fault  the  spring  is  made  of  a  constant 
depth  but  varying  width. 

If  the  spring  is  fixed  at  one  end  and  loaded  at  the 
other  the  plan  should  be  a  triangle  with  the  apex  at 
loaded  end.  If  it  is  supported  at  the  two  ends  and 
loaded  at  the  center,  the  plan  should  be  two  triangles 
with  their  bases  together  under  the  load  forming  a 
6 


82  MACHINE  DESIGN. 

rhombus.     The  deflection  of  such  a  spring  is  one  and 
a  half  times  that  of  a  rectangular  spring. 

As  such  a  spring 
might  be  of  an  in- 
convenient width, 
a  compound  or  leaf- 
spring  is  made  by 
f  cutting  the  trian- 

•       |      I  I       |  — i         gular    spring    into 

^  strips  parallel  to  the 

'   axis,  and  piling  one 

Fig-  24.  above  another  as  in 

Fig.  24. 

This  arrangement  does  not  change  the  principle, 
save  that  the  friction  between  the  leaves  may  increase 
the  resistance  somewhat. 

Let  1= length  of  span. 
b= breadth  of  leaves. 
t= thickness  of  leaves. 
n= number  of  leaves. 
TT^load  at  center. 
A = deflection  at  center. 

S  and  E  may  be  taken  as  80000  and  30000000  re- 
spectively. 
Strength : 

Wl    Snbt2 


(55) 

tj  v 

Elasticity : 


A= 


.(56) 


ELLIPTIC  AND  SEMI-ELLIPTIC  SPEINGS.  83 

35.     Elliptic  and  Semi-Elliptic  Springs.     Springs  as 
they  are  usually  designed  for  service  differ  in  some  re- 
spects from  those 
just  described,  as 
may  be  seen  by  re- 
ference to  Fig.  25. 
A  band  is  used  at 
the  center  to  con- 
fine the  leaves  in  Fig.  25. 
place.  As  this  band 

constrains  the  spring  at  the  center  it  is  best  to  con- 
sider the  latter  as  made  up  of  two  cantilevers  each 

7 /?/1 

having  a  length  of  -^—  where  w  is  the  width  of  band. 

The  spring  usually  contains  several  full-length  leaves 
with  blunt  ends,  the  remaining  leaves  being  graduated 
as  to  length  and  pointed  as  in  Fig.  25.  The  blunt 
full-length  leaves  constitute  cantilevers  of  uniform 
cross-section,  while  the  graduated  leaves  form  canti- 
levers of  uniform  strength.  Under  similar  conditions 
as  to  load  and  fiber  stress  the  latter  will  have  a  deflec- 
tion fifty  per  cent  greater  than  the  former.  Suppos- 
ing that  there  is  no  initial  stress  between  the  leaves 
caused  by  the  band,  both  sets  must  have  the  same  de- 
flection. This  means  that  more  than  its  proportion  of 
the  load  will  be  carried  by  the  full-length  set  and  con- 
sequently it  will  have  a  greater  fiber  stress.  This 
difficulty  can  be  obviated  by  having  an  initial  gap  be- 
tween the  graduated  set  and  the  full-length  set  and 
closing  this  with  the  band. 

If  this  gap  is  made  half  the  working  deflection  of 
the  spring,  the  total  deflection  of  the  graduated  set 
under  the  working  load  will  be  fifty  per  cent  greater 
than  that  of  the  full-length  set  and  the  fiber  stress  will 
be  uniform. 


84:  MACHINE  DESIGN. 

The  load  will  then  be  divided  between  the  two  sets 
in  proportion  to  the  number  of  leaves  in  each. 

One  of  the  full-length  leaves  must  be  counted  as  a 
part  of  the  graduated  set.  When  the  gap  is  closed  by 
a  band  there  will  be  an  initial  pull  on  the  band  due  to 
the  deflection  of  the  spring. 

This  can  be  determined  for  any  given  spring  by  re- 
garding the  two  sets  of  leaves  as  simple  beams  the 
sum  of  whose  deflections  under  the  pull  P  is  equal  to 
the  depth  of  the  gap. 

Full  elliptic  springs  can  be  designed  in  a  similar 
manner  but  the  total  deflection  will  be  double  that  of 
the  semi-elliptic  spring. 

PROBLEMS. 

1.  A  spring  balance  is  to  weigh  25  pounds  with  an  extension 
of  2  inches,  the  diameter  of  spring  being  f  inches  and  the 
material,  tempered  steel. 

Determine  the  diameter  and  length  of  wire,  and  number  of 
coils. 

2.  Determine  the  safe  twisting  moment  and  angle  of  torsion 
for  the  spring  in  example  1,  if  used  for  a  torsional  spring. 

3.  Test  values  of  G  and  S  from  data  given  in  Table  XVIII. 

4.  By  using  above  table  design  a  spring  8  ins.  long  to  carry 
a  load  of  2  tons  without  closing  the  coils  more  than  half  way. 

5.  Design  a  compound  flat  spring  fora  locomotive  to  sustain 
a  load  of  16000  Ibs.  at  the  center,  the  span  being  40  inches,  the 
number  of  leaves  12  and  the  material  steel. 

6.  Determine  the  maximum  deflection  of  the  above  spring, 
under  the  working  load. 

7.  A  semi-elliptic  spring  has  ^V  leaves  in  all  and  n  graduated 
leaves,  and  the  load  on  each  end  is  P=—-.    Develop  formulas 

for  the  fiber  stress  in  each  set  of  leaves  if  there  is  no  initial 
stress. 


ELLIPTIC  AND  SEMI-ELLIPTIC  SPRINGS.  85 

8.  In  Prob.  7  develop  a  formula  for  the  necessary  gap  to 
equalize  the  fiber  stresses. 

9.  In  Prob.  8  determine  the  pull  on  the  band  due  to  the 
initial  stress. 

10.  A  semi-elliptic  spring  has  4  leaves  36  inches  long,  and  12 
graduated  leaves.     The  leaves  are  all   4   inches  wide  and  | 
inches  thick,  and  the  band  at  the  center  is  4  inches  wide.     If 
there  is  no  initial  stress  find  the  share  of  the  load  and  the  fiber 
stress  on  each  set  of  leaves  when  there  is  a  load  of  6  tons  on 
the  center.     Also  determine  deflection. 

11.  In  Prob.   10,   determine  the  amount  of  gap  needed  to 
equalize  the  stresses  in  the  two  sets  of  leaves,  and  the  pull  on 
the  band  at  the  center.     Determine  the  deflection  under  the 
load. 

12.  Measure  various  indicator  springs  and  determine  value 
of  G  from  rating  of  springs. 

13.  Measure  various  brass  extension  springs,  calculate  safe 
static  load  and  safe  stretch. 

14.  Make  an  experiment  on  torsion  spring  to  determine  dis- 
tortion under  a  given  load  and  calculate  value  of  E. 


CHAPTER  VI. 

SLIDING   BEARINGS. 

36.  Slides  in  General.  The  surfaces  of  all  slides 
should  have  sufficient  area  to  limit  the  intensity  of 
pressure  and  prevent  forcing  out  of  the  lubricant.  No 
general  rule  can  be  given  for  the  limit  of  pressure. 
Tool  marks  parallel  to  the  sliding  motion  should  not 
be  allowed,  as  they  tend  to  start  grooving.  The  sliding 
piece  should  be  as  long  as  practicable  to  avoid  local 
wear  on  stationary  piece  and  for  the  same  reason  should 
have  sufficient  stiffness  to  prevent  springing.  A  slide 
which  is  in  continuous  motion  should  lap  over  the 
guides  at  the  ends  of  stroke,  to  prevent  the  wearing  of 
shoulders  on  the  latter  and  the  finished  surfaces  of  all 
slides  should  have  exactly  the  same  width  as  the  sur- 
faces on  which  they  move  for  a  similar  reason. 

Where  there  are  two  parallel  guides  to  motion  as  in 
a  lathe  or  planer  it  is  better  to  have  but  one  of  these 
depended  upon  as  an  accurate  guide  and  to  use  the 
other  merely  as  a  support.  It  must  be  remembered 
that  any  sliding  bearing  is  but  a  copy  of  the  ways  of 
the  machine  on  which  it  was  planed  or  ground  and  in 
turn  may  reproduce  these  same  errors  in  other  ma- 
chines. The  interposition  of  handscraping  is  the  only 
cure  for  these  hereditary  complaints. 

In  designing  a  slide  one  must  consider  whether  it  is 
accuracy  of  motion  that  is  sought,  as  in  the  ways  of 

a  planer  or  lathe,  or  accuracy  of  position  as  in  the  head 

86 


ANGULAR  SLIDES. 


87 


of  a  milling  machine.     Slides  may  be  divided  accord- 
ing to  their  shapes  into  angular,  flat  and  circular  slides. 

37.  Angular  Slides.  An  angular  slide  is  one  in 
which  the  guiding  surface  is  not  normal  to  the  direc- 
tion of  pressure.  There  is  a  tendency  to  displacement 
sideways,  which  necessitates  a  second  guiding  surface 
inclined  to  the  first.  This  oblique  pressure  constitutes 
the  principal  disadvantage  of  angular  slides.  Their 
principal  advantage  is  the  fact  that  they  are  either 
self-adjusting  for  wear,  as  in  the  ways  of  lathes  and 
planers,  or  require  at  most  but  one  adjustment. 

Fig.  26  shows  one  of  the 
V's  of  an  ordinary  planing 
machine.  The  platen  is  held 
in  place  by  gravity.  The 
angle  between  the  two  sur- 
faces is  usually  90  deg.  but 
may  be  more  in  heavy  ma- 
chines. The  grooves  (/,  g  are 
intended  to  hold  the  oil  in 
place ;  oiling  is  sometimes 
effected  by  small  rolls  recessed  into  the  lower  piece 
and  held  against  the  platen  by  springs. 

The  principal  advantage 
of  this  form  of  way  is  its 
ability  to  hold  oil  and  the 
great  disadvantage  its  fa- 
culty for  catching  chips  and 
dirt. 

Fig  27  shows  an  inverted 
V  such  as  is  common  on  the 
ways  of  engine  lathes.  The 
angle  is  about  the  same  as  in  the  preceding  form  but 


Fig.  26. 


Fig.  27 


88 


MACHINE  DESIGN. 


the  top  of  the  V  should  be  rounded  as  a  precaution 
against  nicks  and  bruises. 

The  inverted  V  is  preferred  for  lathes  since  it  will 
not  catch  dirt  and  chips.  It  needs  frequent  lubrica- 
tion as  the  oil  runs  off  rapidly.  Some  lathe  carriages 
are  provided  with  extensions  filled  with  oily  felt  or 
waste  to  protect  the  ways  from  dirt  and  keep  them 
wiped  and  oiled.  Side  pressure  tends  to  throw  the 
carriage  from  the  ways  ;  this  action  may  be  prevented 
by  a  heavy  weight  hung  on  the  carriage  or  by  gibbing 
the  carriage  at  the  back.  (See  Fig.  33).  The  objection 
to  this  latter  form  of  construction  is  the  fact  that  it  is 
practically  impossible  to  make  and  keep  the  two  F's 
and  the  gibbed  slide  all  parallel. 

Fig.  28  shows  a  compound 
V  sometimes  used  on  heavy 
machines.  The  obtuse 
angle  (about  150  deg.)  takes 
the  heavy  vertical  pressure, 
while  the  sides,  inclined 
Fig.  SB.  only  8  or  10  deg.,  take  any 

side  pressure  which    may 
develop. 


38.  Gibbed  Slides.  All  slides  which  are  not  self- 
adjusting  for  wear  must  be  provided  with  gibs  and 
adjusting  screws.  Fig.  29 
shows  the  most  common 
form  as  used  in  tool  slides 
for  lathes  and  planing 
machines. 

The    angle    employed   is 
usually  60  deg. ;  notice  that 


Fig.  29. 

the  corners  c  c  are  clipped  for  strength  and  to  avoid  a 


FLAT  SLIDES. 


89 


corner  bearing  ;  notice  also  the  shape  of  gib.  It  is  bet- 
ter to  have  the  points  of  screws  coned  to  fit  gib  and 
not  to  have  flat  points  fitting  recesses  in  gib.  The 
latter  form  tends  to  spread  joint  apart  by  forcing  gib 
down.  If  the  gib  is  too  thin  it  will  spring  under  the 
screws  and  cause  uneven  wear. 

The  cast  iron  gib,  Fig.  30, 
is  free  from  this  latter  defect 

but  makes  the  slide  rather 

T  /I  clumsy.  The  screws  how- 
ever are  more  accessible  in 
this  form.  Gibs  are  some- 
times made  slightly  tapering 
and  adjusted  by  a  screw  and 


Fig.  30. 


nut  giving  endwise  motion. 

39.  Flat  Slides.  This  type  of  slide  requires  adjust- 
ment in  two  directions  and  is  usually  provided  with 
gibs  and  adjusting  screws.  Flat  ways  on  machine 
tools  are  the  rule  in  English  practice  and  are  gradually 
coming  into  use  in  this  country.  Although  more  ex- 
pensive at  first  and  not  so  simple  they  are  more  durable 
and  usually  more  accurate  than  the  angular  ways. 

Fig.  31  illustrates  a  flat  way  for  a  planing  machine. 
The  other  way  would 
be  similar  to  this  but 
without  adjustment. 
The  normal  pressure 
and  the  friction  are 
less  than  with  angular 
ways  and  no  amount 
of  side  pressure  will 
lift  the  platen  from  its 
position.  Fig.  31. 


90 


MACHINE  DESIGN. 


Fig.   32  shows  a  portion  of  the   ram   of  a  shaping 
machine  and  illustrates  the  use  of  an  L  gih  for  adjust- 

I — . • -.    meiit  in  two  directions. 

•-I 1  Fig.  33  shows  a  gibbed 

slide  for  holding  down 
the  back  of  a  lathe  car- 
riage with  two  adjust- 
ments. 

The   gib  g  is  tapered 
and  adjusted  by  a  screw 
and    nuts.     The    saddle 
of  a  planing  machine  or 
the  table  of  a  shaper  usually  has  a  rectangular  gibbed 


LnJ 


Fig.  33. 


Fig.  34. 


slide  above  and  a  taper  slide  below,  this  form  of  the 
upper  slide  being  necessary  to  hold  the  weight  of  the 
overhanging  metal.  (See  Fig.  34.)  Some  lathes  and 
planers  are  built  with  one  V  or  angular  way  for 
guiding  the  carriage  of  platen  and  one  flat  way  acting 
merely  as  a  support. 


STUFFING  BOXES. 


91 


40.  Circular  Guides.     Examples  of  this  form  may 
be  found  in  the  column  of  the  drill  press  and  the  over- 
hanging   arm  of    the 
The 
steam  ' 


milling  machine. 


Fig.  35. 


cross  heads  of 
engines  are  sometimes 
fitted  with  circular 
guides  ;  they  are  more 
frequently  flat  or  an- 
gular. One  advantage 
of  the  circular  form  is 
the  fact  that  the  cross 
head  can  adjust  itself 
to  bring  the  wrist  pin  parallel  to  the  crank  pin.  The 
guides  can  be  bored  at  the  same  setting  as  the  cylinder 
in  small  engines  and  thus  secure  good  alignment. 

Fig.  35  illustrates  various  shapes  of  cross  head  slides 
in  common  use. 

41.  Stuffing  Boxes.  In  steam  engines  and  pumps 
the  glands  for  holding  the  steam  and  water  packing 
are  the  sliding  bearings  which  cause  the  greatest  fric- 


Fig.  36. 

tion  and  the  most  trouble.     Fig.  36  shows  the  general 


92  MACHINE  DESIGN. 

arrangement.  B  is  the  stuffing  box  attached  to  the 
cylinder  head  ;  R  is  the  piston  rod  ;  G  the  gland  ad- 
justed by  nuts  on  the  studs  shown  ;  P  the  packing 
contained  in  a  recess  in  the  box  and  consisting  of  rings, 
either  of  some  elastic  fibrous  material  like  hemp  and 
woven  rubber  cloth  or  of  some  soft  metal  like  babbit. 
The  pressure  between  the  packing  and  the  rod,  neces- 
sary to  prevent  leakage  of  steam  or  water,  is  the  cause 
of  considerable  friction  and  lost  work.  Experiments 
made  from  time  to  time  in  the  laboratories  of  the  Case 
School  have  shown  the  extent  and  manner  of  variation 
of  this  friction.  The  results  for  steam  packings  may 
be  summarized  as  follows : 

1.  That  the  softer  rubber  and   graphite   packings, 
which  are  self-adjusting  and  self-lubricating,   as  in 
Nos.  2,  3,  7,  8,  and  11,  consume  less  power  than  the 
harder  varieties.     No.  17,  the  old  braided  flax  style, 
gives  very  good  results. 

2.  That  oiling  the  rod  will  reduce  the  friction  with 
any  packing. 

3.  That  there  is  almost  no  limit  to  the  loss  caused 
by  the  injudicious  use  of  the  monkey-wrench. 

4.  That  the  power  loss  varies  almost  directly  with 
the  steam  pressure  in  the  harder  varieties,  while  it  is 
approximately  constant  with  the  softer  kinds. 

The  diameter  of  rod  used — two  inches — would  be  ap- 
propriate for  engines  from  50  to  100  horse-power.  The 
piston  speed  was  about  140  feet  per  minute  in  the 
experiments,  and  the  horse-power  varied  from  .036  to 
.400  at  50  pounds  steam  pressure,  with  a  safe  average 
for  the  softer  class  of  packings  of  .07  horse- 
power. 

At  a  piston  speed  of  600  feet  per  minute,  the  same 
friction  would  give  a  loss  of  from  .154  to  1.71  with  a 


STUFFING  BOXES. 


93 


working  average  of  .30  horse-power,  at  a  mean  steam 
pressure  of  50  pounds. 

In  Table  XIX  Nos.  6,  14,  15  and  16  are  square,  hard 
rubber  packings  without  lubricants. 

Similar  experiments  on  hydraulic  packings  under 
a  water  pressure  varying  from  ten  to  eighty  pounds 
per  square  inch  gave  results  as  shown  in  Table 
XXI. 

The  figures  given  are  for  a  two  inch  rod  running 
at  an  average  piston  speed  of  140  feet  per  minute. 

TABLE  XIX. 


0) 

•gji 

o5 

S*S  8Q 

gg^S 

°2  i 

if 

H 

-Jl 

®  S^ 

0?    02 

Remarks  on  Leakage,  etc. 

MS 

.1 

J-s'S 

P4         * 

o  o  ,0 

1 

5 

22 

.091 

.085 

Moderate  leakage. 

2 
3 

8 
5 

40 
25 

.049 
.037 

.048 
.036 

Easily  adjusted  ;  slight  leakage. 
Considerable  leakage. 

4 

5 

25 

.159 

.176 

Leaked  badly. 

5 

5 

25 

.095 

.081 

Oiling  necessary  ;  leaked  badly. 

6 

5 

25 

.368 

.400 

Moderate  leakage. 

7 

5 

25 

.067 

.067 

Easily  adjusted  and  no  leakage. 

8 

5 

25 

.082 

.082 

Very  satisfactory  ;  slight  leakage. 

9 

3 

15 

.200 

.182 

Moderate  leakage. 

10 

3 

.275 

Excessive  leakage. 

11 

5 

25 

.157 

.172 

Moderate  leakage. 

12 

5 

25 

.266 

.330 

Moderate  leakage. 

13 

5 

25 

.162 

.230 

No  leakage  ;  oiling  necessary. 

14 

5 

25 

.176 

.276 

Moderate  leakage  ;  oiling  necessary. 

15 

5 

25 

.233 

.255 

Difficult  to  adjust  ;  no  leakage. 

16 

5 

25 

.292 

.210 

Oiling  necessary  ;  no  leakage. 

17 

5 

25 

.128 

.084 

No  leakage. 

MACHINE  DESIGN. 
TABLE  XX. 


be 
B 

3 

B 
S 

Horse  Power  consumed  by  each  Box,  when 
Pressure  was  applied  to  Gland  Nuts 
by  a  Seven-Inch  Wrench. 

53^  fco 

£  c.S 

O  &  72 
O   Q)  *^ 

5 

Pounds. 

8 

Pounds. 

10 

Pounds. 

12 

Pounds. 

14 

Pounds. 

16 

Pounds. 

Dry. 

Oiled. 

1 
3 
4 
5 
6 
7 
8 
9 
11 
12 
13 
15 
16 
17 

.120 



.136 

.055 
.154 

.021 
.123 

.... 

.248 
.220 

... 

.303 

.... 

.390 

.... 

.348 
.126 
.363 
.666 
.405 
.161 
.317 
.526 
.327 
.198 

.430 

.228 
.500 

.323 
.067 
.533 
.666 
.454 
.454 

.194 
.053 
.236 
.636 
.176 
.122 

.260 
.535 

.330 
.520 

.340 
.533 

.454 
.242 
.394 

'.860 

.277 

.359 

.582 

.454 

.... 

.380 

TABLE  XXI. 


No.  of 

Av.  H.  P. 

Av.  H.  P. 

Max. 

Min. 

Av.  H.  P. 

at 

at 

for  entire 

Packing. 

20  Ibs. 

70  Lbs. 

H.   P. 

H.   P. 

Test. 

1 

.077 

.351 

.452 

.024 

.259 

2 

.422 

.500 

.512 

.167 

.410 

3 

.130 

.178 

.276 

.035 

.120 

4 

.184 

.195 

.230 

.142 

.188 

5 

.146 

.162 

.285 

.069 

.158 

6 

.240 

.200 

.255 

.071 

.186 

7 

.127 

.192 

.213 

.095 

.154 

8 

.153 

.174 

.238 

.112 

.165 

9 

.287 

.469 

.535 

.159 

.389 

10 

.151 

.160 

.226 

.035 

.103 

11 

.141 

.156 

.380 

.064 

.177 

12 

.053 

.095 

.143 

.035 

.090 

STUFFING  BOXES.  95 

Packings  Nos.  5,  6,  10  and  12  are  braided  flax  with 
graphite  lubrication  and  are  best  adapted  for  low 
pressures.  Packings  Nos.  3,  4  and  7  are  similar  to  the 
above  but  have  paraffine  lubrication.  Packings  Nos. 
2  and  9  are  square  duck  without  lubricant  and  are  only 
suitable  for  very  high  pressures,  the  friction  loss  being 
approximately  constant. 

PROBLEMS. 

Make  a  careful  study  and  sketch  of  the  sliding  bearings  on 
each  of  the  following  machines  and  analyze  as  to  (a)  Purpose 
(6)  Character,  (c)  Adjustment,  (d)  Lubrication. 

1.  An  engine  lathe. 

2.  A  planing  machine. 

3.  A  shaping  machine. 

4.  A  milling  machine. 

5.  An  upright  drill. 

6.  A  Corliss  engine. 

7.  A  Porter- Allen  engine. 

8.  A  gas-engine. 

9.  An  air-compressor. 


CHAPTER  VII. 


JOURNALS,  PIVOTS  AND  BEARINGS. 

42.  Journals.     A  journal  is  that  part  of  a  rotating 
shaft  which  rests  in  the  bearings  and  is  of  necessity  a 
surface  of  revolution,  usually  cylindrical  or  conical. 
The  material  of  the  journal  is  generally  steel,  some- 
times soft  and  sometimes  hardened  and  ground. 

The  material  of  the  bearing  should  be  softer  than 
the  journal  and  of  such  a  quality  as  to  hold  oil  readily. 
The  cast  metals  such  as  cast  iron,  bronze  and  babbit 
metal  are  suitable  on  account  of  their  porous,  granular 
character.  Wood,  having  the  grain  normal  to  the 
bearing  surface,  is  used  where  water  is  the  lubricant, 
as  in  water  wheel  steps  and  stern  bearings  of  propellers. 

43.  Adjustment.     Bearings  wear  more  or  less  rapidly 
with  use  and  need  to  be  adjusted  to  compensate  for 

the  wear.  The  adjust- 
ment must  be  of  such  a 
character  and  in  such  a 
direction  as  to  take  up  the 
wear  and  at  the  same 
time  maintain  as  far  as 
possible  the  correct  shape 
of  the  bearing.  The  ad- 
justment should  then  be 
in  the  line  of  the  greatest 
pressure. 

Fig.  37  illustrates  some 


Fig.  37. 


of  the  more  common  ways  of  adjusting  a  bearing,  the 

96 


ADJUSTMENT. 


97 


arrows  showing  the  direction  of  adjustment  and  presu- 
mably the  direction  of  pressure  ;  (a)  is  the  most  usual 
where  the  principal  wear  is  -vertical,  (d)  is  a  form 
frequently  used  on  the  main  journals  of  engines  when 
the  wear  is  in  two  directions,  horizontal  on  account  of 
the  steam  pressure  and  vertical  on  account  of  the 
weight  of  shaft  and  fly  wheel.  All  of  these  are  "more 
or  less  imperfect  since  the  bearing,  after  wear  and  ad- 
justment, is  no  longer  cylindrical  but  is  made  up  of 
two  or  more  approximately  cylindrical  surfaces, 

A  bearing  slightly 
conical  and  adjusted 
endwise  as  it  wears,  is 
probably  the  closest 
approximation  to  cor- 
rect practice. 

Fig.  38  shows  the 
main  bearing  of  the 

Porter  -  Allen    engine,  N — — — ' 

one    of    the    best   ex- 
amples of  a  four  part  adjustment.     The  cap,  is  adjusted 

in  the  normal  way  with 
bolts  and  nuts  ;  the  bot- 
tom, can  be  raised  and 
lowered  by  liners  placed 
underneath  ;  the  cheeks 
can  be  moved  in  or 
out  by  means  of  the 
wed  ges  shown.  Thus  it 
is  possible,  not  only  to 
adjust  the  bearing  for 
wear,  but  to  align  the 
shaft  perfectly. 


Fig.  39. 


A  three  part  bearing 


98 


MACHINE  DESIGN. 


for  the  main  journal  of  an  engine  is  shown  in  Fig. 
39.  In  this  bearing  there  is  one  horizontal  adjust- 
ment, instead  of  two  as  in  Fig.  38. 

The  main  bearing  of  the 
spindle  in  a  lathe,  as  shown  in 
Fig.  40,  offers  a  good  example 
1  of  symmetrical  adjustment. 
f\  The  headstock  A  has  a  conical 
U  hole  to  receive  the  bearing  B, 
which  latter  can  be  moved 
lengthwise  by  the  nuts  F  G* 
The  bearing  may  be  split  into 
two,  three  or  four  segments  or 
it  may  be  cut  as  shown  in  (e) 
Fig.  37,  and  sprung  into  adjustment,  A  careful 
distinction  must  be  made  between  this  class  of  bearing 
and  that  before  mentioned,  where  the  journal  itself  is 
conical  and  adjusted  endwise.  A  good  example  of 
the  latter  form  is  seen  in  the  spindles  of  many  milling 
machines. 


Fig.  40. 


Fig.  41. 

Fig.  41  shows  the  spindle  of  an  engine  lathe  complete 
with  its  two  bearings.  The  end  thrust  is  taken  by  a 
fiber  washer  backed  by  an  adjusting  collar  and  check 
nut.  Both  bearings  belong  to  the  class  shown  in 
Fig.  40. 

A  conical  journal  with  end  adjustment  is  illustrated 


LUBRICATION.  99 

in  Fig,   42,    which  shows   the    spindle  of  a  milling 
machine*     The  front  journal  is  conical  and  is  adjusted 


Fig.  42. 

for  wear  by  drawing  it  back  into  its  bearing  with  the 
nut.  The  rear  journal  on  the  other  hand  is  cylindrical 
and  its  bearing  is  adjusted  as  are  those  just  described. 
The  end  thrust  is  taken  by  two  loose  rings  at  the  front 
end  of  the  spindle, 

44.  Lubrication.  The  bearings  of  machines  which 
run  intermittently,  like  most  machine  tools,  are  oiled 
by  means  of  simple  oil  holes,  but  machinery  which  is 
in  continuous  motion  as  is  the  case  with  line  shafting 
and  engines  requires  some  automatic  system  of  lubri- 
cation. There  is  not  space  in  this  book  for  a  detailed 
description  of  all  the  various  types  of  oiling  devices 
and  only  a  general  classification  will  be  attempted. 

Lubrication  is  effected  in  the  following  ways  : 

1.  By  grease  cups, 

2.  By  oil  cups. 

3.  By  oily  pads  of  felt  or  waste. 

4.  By  oil  wells  with  rings  or  chains  for  lifting  the 
oil. 

5.  By  centrifugal  force  through  a  hole  in  the  journal 
itself. 

Grease  cups  have  little  to  recommend  them  except 


100 


MACHINE  DESIGN. 


as  auxiliary  safety  devices.  Oil  cups  are  various  in 
their  shapes  and  methods  of  operation  and  constitute 
the  cheap  class  of  lubricating  devices.  They  may  be 
divided  according  to  their  operation  into  wick  oilers, 
needle  feed,  and  sight  feed.  The  two  first  mentioned 
are  nearly  obsolete  and  the  sight  feed  oil  cup,  which 
drops  the  oil  at  regular  intervals  through  a  glass  tube 
in  plain  sight,  is  in  common  use.  The  best  sight  feed 
oiler  is  that  which  can  be  readily  adjusted  as  to  time 
intervals,  which  can  be  turned  on  or  off  without  dis- 
turbing the  adjustment  and  which  shows  clearly  by  its 
appearance  whether  it  is  turned  on.  On  engines  and 
electric  machinery  which  is  in  continuous  use  day  and 
night,  it  is  very  important  that  the  oiler  itself  should 
be  stationary,  so  that  it  may  be  filled  without  stopping 
the  machinery. 

A  modern  sight  feed  oiler  for  an  engine  is  illustrated 
in  Fig.  43.  T  is  the  glass  tube  where  the  oil  drop 
is  seen.  The  feed  is  regulated  by  the 
nut  N,  while  the  lever  L  shuts  off  the 
oil.  Where  the  lever  is  as  shown  the 
oil  is  dropping,  when  horizontal  the 
oil  is  shut  off. 

The  nut  can  be  adjusted  once  for 
all,  and  the  position  of  the  lever  shows 
immediately  whether  or  not  the  cup  is 
in  use. 

In  modern  engines  particular  at- 
tention has  been  paid  to  the  problem 
of  continuous  oiling.  The  oil  cups  are 
all  stationary  and  various  ingenious 
devices  are  used  for  catching  the  drops 
of  oil  from  the  cups  and  distributing 
them  to  the  bearing  surfaces, 


LUBRICATION. 


101 


For  continuous  oiling  of  stationary  bearings  as  in 
line  shafting  and  electric  machinery,  an  oil  well  below 
the  bearing  is  preferred,  with  some  automatic  means 
of  pumping  the  oil  over  the  bearing,  when  it  runs 
back  by  gravity  into  the  well.  Por- 
ous wicks  and  pads  acting  by  capil- 
lary attraction  are  uncertain  in 
their  action  and  liable  to  become 
clogged.  For  bearings  of  medium 
size,  one  or  more  light  steel  rings 
running  loose  on  the  shaft  and  dip- 
ping into  the  oil,  as  shown  in  Fig.  44, 
are  the  best.  For  large  bearings 
flexible  chains  are  employed  which 
take  up  less  room  than  the  ring.  Centrifugal  oilers 
are  most  used  on  parts  which  cannot  readily  be  oiled 
when  in  motion,  such  as  loose  pulleys  and  the  crank 
pins  of  engines. 

Fig.  45  shows  two  such  devices  as  applied  to  an  en- 
gine. In  A  the  oil  is  supplied  by  the  waste  from  the 
main  journal ;  in  B  an  external  sight-feed  oil  cup  is 
used  which  supplies  oil  to  the  central  revolving  cup  (7. 


Fig.  44. 


Fig.  45. 

Loose  pulleys  or  pulleys  running  on  stationary  studs 
are  best  oiled  from  a  hole  running  along  the  axis 


102 


MACHINE  DESIGN. 


of  the  shaft  and  thence  out  radially  to  the  surface  of 
the  bearing.  See  Fig  46.  A  loose  bushing  of  some 
soft  metal  perforated  with  holes  is  a  good  safety  device 
for  loose  pulleys. 


Fig.  46. 

Note  :  For  adjustable  pedestal  and  hanging  bear- 
ings see  the  chapter  on  shafting. 

45:    Friction  of  Journals  : 
Let  T^=the  total  load  of  a  journal  in  Ibs. 
Z=the  length  of  journal  in  inches. 
c?=the  diameter  of  journal  in  inches. 
N=  number  of  revolutions  per  minute. 
v= velocity  of  rubbing  in  feet  per  minute. 
F=  friction  at  surface  of  journal  in  Ibs. 

=  W  tan  ^  nearly,  whose  ^  is  the  angle  of  re- 
pose for  the  two  materials. 

If  a  journal  is  properly  fitted  in  its  bearing  and  does 
not  bind,  the  value  of  F  will  not  exceed  W  tan  $  and 
may  be  slightly  less.  The  value  of  tan  \l/  varies  accord- 
ing to  the  materials  used  and  the  kind  of  lubrication, 
from  .05  to  .01  or  even  less.  See  experiments  described 
in  Art.  48.  The  work  absorbed  in  friction  may  be  thus 
expressed  : 

.  per  min.  (57) 


LIMITS  OF  PRESSURE.  103 

46.  Limits  of  Pressure.  Too  great  an  intensity  of 
pressure  between  the  surface  of  a  journal  and  its  bear- 
ing will  force  out  the  lubricant  and  cause  heating 
and  possibly  "  seizing."  The  safe  limit  of  pressure  de- 
pends on  the  kind  of  lubricant,  the  manner  of  its  ap- 
plication and  upon  whether  the  pressure  is  continuous 
or  intermittent.  The  projected  area  of  a  journal,  or 
the  product  of  its  length  by  its  diameter,  is  used  as  a 
divisor. 

The  journals  of  railway  cars  offer  a  good  example 
of  continuous  pressure  and  severe  service.  A  limit  of 
300  pounds  per  square  inch  of  projected  area  has  been 
generally  adopted  in  such  cases. 

In  the  crank  and  wrist  pins  of  engines,  the  reversal 
of  pressure  diminishes  the  chances  of  the  lubricant 
being  squeezed  out,  and  a  pressure  of  500  Ibs.  per 
sq.  in.  is  generally  allowed. 

The  use  of  heavy  oils  or  of  an  oil  bath,  and  the  em- 
ployment of  harder  materials  for  the  journal  and  its 
bearing  allow  of  even  greater  pressures. 

Professor  Barr's  investigations  of  steam  engine  pro- 
portions *  show  that  the  pressure  per  square  inch  on 
the  cross-head  pin  varies  from  ten  to  twenty  times  that 
on  the  piston,  while  the  intensity  of  pressure  on  the 
crank  pin  is  from  two  to  eight  times  that  on  the  piston. 
Allowing  a  mean  pressure  on  the  piston  of  fifty  pounds 
per  square  inch  would  give  the  following  range  of 
pressures  : 

Minimum.     Maximum. 

Wrist  pins.  .         .  500  1000 

Crank  pins.          .         .  100  400 

The  larger  values  for  the  wrist  pins  are  allowable  on 

*  Trans.  A.  S.  M.  E.,  Vol.  XVIII. 


104  MACHINE  DESIGN. 

account  of  the  comparatively  low  velocity  of  rubbing. 
Naturally  the  larger  values  for  the  pressure  are  found 
in  the  low  speed  engines. 

A  discussion  of  the  subject  of  bearings  is  reported  in 
the  transactions  of  the  American  Society  of  Mechanical 
Engineers  for  1905-06  *  and  some  valuable  data  are 
furnished.  Mr.  Geo.  M.  Basford  says  that  locomotive 
crank  pins  have  been  loaded  as  high  as  1500  to  1700 
pounds  per  square  inch,  and  wrist  pins  to  4000  pounds 
per  square  inch. 

Locomotive  driving  journals  on  the  other  hand  are 
limited  to  the  following  pressures  : 

Passenger  locomotives     .     190  Ibs.  per  sq.  in. 
Freight    .  "  .200 

Switching  "  .     220 

Cars  and  tender  bearings.     300 

Mr.  H.  G.  Eeist  gives  some  figures  on  the  practice 
of  the  General  Electric  Company,  for  motors  and 
generators. 

This  company  allows  from  30  to  80  pounds  pressure 
per  square  inch  with  an  average  value  of  from  40  to  45 
pounds.  The  rubbing  speeds  vary  from  40  feet  to  1200 
feet  per  minute.  Mr.  Eeist  quotes  approvingly  the 
formula  of  Dr.  Thurston's,  viz  :  That  the  product  of 
the  pressure  in  pounds  per  square  inch  and  the  rubbing 
speed  in  feet  per  minute  should  not  exceed  50,000. 

A  careful  reading  of  the  whole  discussion  will  repay 
any  one  who  has  to  design  shaft  bearings  of  any  de- 
scription. 

47.  Heating  of  Journals.  The  proper  length  of 
journals  depends  on  the  liability  of  heating. 

*  Trans.  A.  S.  M.  E.,  Vol.  XXVII. 


HEATING  OF  JOURNALS.  105 

The  energy  or  work  expended  in  overcoming  friction 
is  converted  into  heat  and  must  be  conveyed  away  by 
the  material  of  the  rubbing  surfaces.  If  the  ratio  of 
this  energy  to  the  area  of  the  surface  exceeds  a  certain 
limit,  depending  on  circumstances,  the  heat  will  not 
be  conveyed  away  with  sufficient  rapidity  and  the 
bearing  will  heat. 

The  area  of  the  rubbing  surface  is  proportional  to 
the  projected  area  or  product  of  the  length  and  diame- 
ter of  the  journal,  and  it  is  this  latter  area  which  is 
used  in  calculation. 

Adopting  the  same  notation  as  is  used  in  Art.  45, 
we  have  from  equation  (57). 


the  work  of  friction  =.   ft.  ibs.  per  min. 

\2i 

or  =TrdNWtan\t>  inch  Ibs. 
The  work  per  square  inch  of  projected  area  is  then  : 


ld 
Solving  in  (a)  for  Z 


Z=!l±LJLl±!±lir.  (b) 

w 

Let  —2 — — =  C  a  co-efficient  whose  value  is  to  be  ob- 
tained by  experiment ;  then 

C= — j —  and  /=    ~  -.        .        .        (58) 

Crank  pins  of  steam  engines  have  perhaps  caused 
more  trouble  by  heating  than  any  other  form  of  jour- 
nal. A  comparison  of  eight  different  classes  of  propel- 
lers in  the  old  U.  S.  Navy  showed  an  average  value 
for  C  of  350,000. 


106  MACHINE  DESIGN. 

A  similar  average  for  the  crank  pins  of  thirteen 
screw  steamers  in  the  French  Navy  gave  (7=400,000. 

Locomotive  crank  pins  which  are  in  rapid  motion 
through  the  cool  outside  air  allow  a  much  larger  value 
of  (7,  sometimes  more  than  a  million. 

Examination  of  ten  modern  stationary  engines 
shows  an  average  value  of  (7=200,000  and  an  average 
pressure  per  square  inch  of  projected  area  =300  Ib. 

The  investigations  of  Professor  Barr  above  referred 
to  show  a  wide  variation  in  the  constants  for  the  length 
of  crank  pins  in  stationary  engines.  He  prefers  to  use 

TJ"D 

the  formula  :   l=K-T-+B  where   K  and  B  are  con- 

JL/ 

stants  and  L=  length  of  stroke  of  engine  in  inches. 
We  may  put  this  in  another  form  since  : 

HP      WN 

-T-=—       -  where  Wis  the  total  mean  pressure. 
JL/      lyoUuu 

The  formula  then  becomes  : 


The  value  of  B  was  found  to  be  2.5  in.  for  high- 
speed and  2  in.  for  low-speed  engines,  while  K  fluc- 
tuated from  .13  to  .46  with  an  average  of  .30  in  the 
the  former  class,  and  from  .40  to  .80  with  an  average 
of  .60  in  the  low-speed  engines. 

If  we  adopt  average  values  we  have  the  following 
formulas  for  the  crank-pins  of  modern  stationary 
engines  : 

WN 
High-speed  engines  Z=-+  2.  5  in.      .      . 


WN 
Low-speed   engines  1=  +2  in  .....  (61) 


EXPERIMENTS.  107 

Compare  these  formulas  with  (58)  when  values  of  C 
are  introduced. 

In  a  discussion  on  the  subject  of  journal  bearings  in 
1885,*  Mr.  Geo.  H.  Babcock  said  that  he  had  found  it 
practicable  to  allow  as  high  as  1200  Ib.  per  sq.  in.  on 
crank  pins  while  the  main  journal  could  not  carry 
over  300  Ib.  per  sq.  in.  without  heating.  One  rule 
for  speed  and  pressure  of  journal  bearings  used  by  a 
well-known  designer  of  Corliss  engines  is  to  multiply 
the  square  root  of  the  speed  in  feet  per  second  by  the 
pressure  per  square  inch  of  projected  area  and  limit 
this  product  to  350  for  horizontal  engines  and  500  in 
vertical  engines. 

48.  Experiments.  Some  tests  made  on  a  steel  journal 
3^  inches  in  diameter  and  8  inches  long  running  in  a 
cast-iron  bearing  and  lubricated  by  a  sight-feed  oiler, 
will  serve  to  illustrate  the  friction  and  heating  of  such 
journals. 

The  two  halves  of  the  bearing  were  forced  together 
by  helical  springs  with  a  total  force  of  1400  pounds,  so 
that  there  was  a  pressure  of  54  Ib.  per  sq.  in.  on  each 
half.  The  surface  speed  was  430  ft.  per  min.  and  the 
oil  was  fed  at  the  rate  of  about  12  drops  per  minute. 
The  lubricant  used  was  a  rather  heavy  automobile  oil 
having  a  specific  gravity  of  0.925  and  a  viscosity  of 
174  when  compared  with  water  at  20  deg.  Cent. 

The  length  of  the  run  was  two  hours  and  the  tem- 
perature of  the  room  70  deg.  Fahr.  (See  Table  XXII.) 

*  Trans.  A.  S.  M.  E.,  Vol.  VI. 


108 


MACHINE  DESIGN. 

TABLE  XXII. 

FRICTION  OF  JOURNAL  BEARING. 


Time. 

Rev.  per  min. 

Temp.  Fahr. 

Coeff.  of  friction. 

10:03 

500    • 

69 

.024 

10:15 

482 

82 

.0175 

10:30 

506 

100 

.013 

10:45 

506 

115 

.010 

11:00 

516 

125 

.010 

11  :15 

135 

.004 

11  :30 

145 

.004 

11  :45 

512 

147 

.004 

12:00 



151 

.007 

49-  Strength  and  Stiffness  of  Journals.     A  journal 
is  usually  in  the  condition  of  a  bracket  with  a  uniform 

load,  and  the  bending  moment  M =—&- 
Therefore  by  formula  (6) 

\.\Wl 


The  maximum  deflection  of  such  a  bracket  is 

A      W? 

~SE  I 


64  WV 


If  as  is  usual  A  is  allowed  to  be 
stiffness  d- 


or  approximately    d=4.J 


E 

*IWF 
E   ' 


inches,  then  for 
(63) 

.(64) 


CAPS  AND  BOLTS.  109 

The  designer  must  be  guided  by  circumstances  in 
determining  whether  the  journal  shall  be  calculated 
for  wear,  for  strength  or  for  stiffness.  A  safe  way  is 
to  design  the  journal  by  the  formulas  for  heating  and 
wear  and  then  to  test  for  strength  and  deflection. 

Kemember  that  no  factor  of  safety  is  needed  in 
formula  for  stiffness. 

Note  that  W  in  formulas  for  strength  and  stiffness 
is  not  the  average  but  the  maximum  load. 

50.  Caps  and  Bolts.  The  cap  of  a  journal  bearing 
exposed  to  upward  pressure  is  in  the  condition  of  a 
beam  supported  by  the  holding  down  bolts  and  loaded 
at  the  center,  and  may  be  designed  either  for  strength 
or  for  stiffness. 
Let :  P=max.  upward  pressure  on  cap. 

L= distance  between  bolts. 

b= breadth  of  cap  at  center. 

h= depth  of  cap  at  center. 

A = greatest  allowable  deflection. 

Strength:      M-™* 


Stiffness: 


•25ZT (65) 


T_bh'_WL* 
12  ~ 


.(66) 


If  A  is  allowed  to  be  TJT  inches  and  E  for  cast  iron  is 
taken  =18000000 


then:  &».01115Zr~f-7- (67) 


110  MACHINE  DESIGN. 

The  holding  down  bolts  should  be  so  designed  that 
the  bolts  on  one  side  of  the  cap  may  be  capable  of 
carrying  safely  two  thirds  of  the  total  pressure. 

PROBLEMS. 

1 .  A  flat  car  weighs  10  tons,  is  designed  to  carry  a  load  of 
20  tons  more  and  is  supported  by  two  four-wheeled  trucks,  the 
axle  journals  being  of  wrought  iron  and  the  wheels  33  inches 
in  diameter. 

Design  the  journals,  considering  heating,  wear,  strength 
and  stiffness,  assuming  a  maximum  speed  of  30  miles  an  hour, 
factor  of  safety =10  and  C=  300000. 

2.  The  following  dimensions  are  those  generally  used  for  the 
journals  of  freight  cars  having  nominal  capacities  as  indicated : 

CAPACITY.  DIMENSIONS  OF  JOURNAL. 

100000  lb.  4. 5  by  9  in. 

60000  lb.  4.25  by  8  in. 

40000  lb.  3.75  by  7  in. 

Assuming  the  weight  of  the  car  to  be  40  per  cent  of  its 
carrying  capacity  in  each  instance,  determine  the  pressure  per 
square  inch  of  projected  area  and  the  value  of  the  constant  C 
{Formula  (58)} . 

3.  Measure  the  crank  pin  of  any  modern  engine  which  is 
accessible,  calculate  the  various  constants  and  compare  them 
with  those  given  in  this  chapter. 

4.  Design  a  crank  pin  for  an  engine  under  the  following  con- 
ditions : 

Diameter  of  piston  =28  inches. 

Maximum  steam  pressui-e=90  lb.  per  sq.  in. 
Mean  steam  pressure  =40  lb.  per  sq.  in. 
Revolutions  per  minute  =75 

Determine  dimensions  necessary  to  prevent  wear  and  heating 
and  then  test  for  strength  and  stiffness. 

5.  Design  a  crank  pin  for  a  high  speed  engine  having  the 
following  dimensions  and  conditions  : 

Diameter  of  piston  =14  inches 


STEP-BEARINGS.  HI 

Maximum  steam  pressure =100  Ib.  per  sq.  in. 
Mean  steam  pressure         =50  Ib.  per  sq.  in. 
Revolutions    per    minute=250. 

6.  Make  a  careful  study  and  sketch  of  journals  and  journal 
bearings  on  each  of  the  following  machines  and  analyze  as  to 
(a)  Materials,     (b)  Adjustment,     (c)  Lubrication. 

a.  An  engine  lathe. 

b.  A  milling  machine. 

c.  A  steam  engine. 

d.  An  electric  generator  or  motor. 

7.  Sketch  at  least  two  forms  of  oil  cup  used  in  the  labora- 
tories and  explain  their  working. 

8.  The  shaft  journal  of  a  vertical  engine  is  4  in.  in  diameter 
by  6  in.  long.     The  cap  is  of  cast  iron,   held  down  by  4  bolts 
of  wrought  iron,  each  5   in.  from   center  of  shaft,   and  the 
greatest  vertical  pressure  is  12000  Ib. 

Calculate  depth  of  cap  at  center  for  both  strength  and  stiff- 
nesSi  and  also  the  diameter  of  bolts. 

9.  Investigate  the  strength   of  the   cap  and  bolts  of  some 
pillow  block  whose  dimensions  are  known,  under  a  pressure  of 
500  Ib.  per  sq.  in.  of  projected  area. 

10.  The  total  weight  on    the  drivers    of  a  locomotive  is 
64000   Ib.     The   drivers   are  four   in  number,    5  ft.   2  in.    in 
diameter,  and  have  journals  7£  in.  in  diameter. 

Determine  horse  power  consumed  in  friction  when  the 
locomotive  is  running  50  miles  an  hour,  assuming  tan</>=.05. 

51.  Step-Bearings.  Any  bearing  which  is  designed 
to  resist  end  thrust  of  the  shaft  rather  than  lateral 
pressure  is  denominated  a  step  or  thrust  bearing. 
These  are  naturally  most  used  on  vertical  shafts,  but 
may  be  frequently  seen  on  horizontal  ones  as  for  ex- 
ample on  the  spindles  of  engine  lathes,  boring  machines 
and  milling  machines. 

Step-bearings  may  be  classified  according  to  the 
shape  of  the  rubbing  surface,  as  flat  pivots  and  collars, 
conical  pivots,  and  conoidal  pivots  of  which  the  Schiele 


112  MACHINE  DESIGN. 

pivot  is  the  best  known.  When  a  step-bearing  on  a 
vertical  shaft  is  exposed  to  great  pressure  or  speed  it 
is  sometimes  lubricated  by  an  oil  tube  coming  up  from 
below  to  the  center  of  the  bearing  and  connecting  with 
a  stand  pipe  or  force-pump.  The  oil  entering  at  the 
center  is  distributed  by  centrifugal  force. 

52.      Friction   of   Pivots  or  Step-bearings.  —  Flat 
Pivots. 

Let  W--  weight  on  pivot 

di=-.outer  diameter  of  pivot 

p=  intensity  of  vertical  pressure 

T=  moment  of  friction 

/=  co-efficient  of  friction  —tan  <f> 

We  will  assume  p  to  be  a  constant  which  is  no  doubt 
approximately  true. 


area 


Let  r=the  radius  of  any  elementary  ring  of  a  width 
=  c?r,  then  area  of  element  =2irrdr 

Friction   of   element  =  fp  X^ 
Moment  of  friction  of  element  =  2fpirr*dr 


and  T=2fp7r  i  ^-r2dr (a) 


or    - 


,„„,. 

(68) 


The  great  objection  to  this  form  of  pivot  is  the  un- 
even wear  due  to  the  difference  in  velocity  between 
and  circumferencef 


CONICAL  PIVOT.  113 

53.  Flat  Collar. 

Let  d  2= inside  diameter 
Integrating  as   in  equation   (a)  above,    but   using 

limits  — l and— 2  we  have 

2i  2i 


In  this  case 

P= 

and  T=lWJ%£=%j.  .     .     (69) 


54.  Conical  Pivot. 

Let  a = angle  of  inclination  to  the  vertical. 


VJ 


P= 
7  dW      dP 


As  in  the  case  of  a  flat 
ring  the  intensity  of  the 
vertical  pressure  is 

4:W 


\ 


/and  the  vertical  pressure 
on  an  elementary  ring  of 
*  the  bearing  surface  is 

Fig.  47. 


\ 


dW=-^^rdr=^ 

-C*«J  Cti  —  1*2 


As  seen  in  Fig.    4T   the  normal  pressure  on  the 
elementary  ring  is 

flp_dW        SWrdr 
3 


MACHINE  DESIGN. 

The  friction  on  the  ring  is  fdP  and  the  moment  of 
this  friction  is 


(dl— 
J  - 


(70) 


As  a  approaches  |  the  value  of  T  approaches  that  of  a 

flat  ring,  and  as  a  approaches  0  the  value  of  T  ap- 
proaches QO  . 
If  d2=0  we  have 


T  = 

The  conical  pivot  also  wears  unevenly,  usually  as- 
suming a  concave  shape  as  seen  in  profile. 

55.  Schiele's  Pivot.  By  experimenting  with  a 
pivot  and  bearing  made  of  some  friable  material,  it  was 
shown  that  the  outline  tended  to  become  curved  as 
shown  in  Fig.  49.  This  led  to  a  mathematical  investi- 
gation which  showed  that  the  curve  would  be  a  trac- 
trix  under  certain  conditions. 

This  curve  may  be  traced  me- 
chanically  as  shown  in  Fig.  48. 

Let  the  weight  W  be  free  to 
move  on  a  plane.  Let  the  string 
SW  be  kept  taut  and  the  end  S 
moved  along  the  straight  line  SL. 
Then  will  a  pencil  attached  to  the 
F.  4g  center  of  W  trace  on  the  plane  a 

tractrix  whose  axis  is  SL, 


SCHIELE'S  PIVOT. 


115 


In  Fig.  49  let  SW= length 
of  string  =r,  and  let  P  be  any 
point  in  the  curve.  Then  it  is 
evident  that  the  tangent  PQ  to 
the  curve  is  a  constant  and  =rl 

Also  -t-»=i\ 
sinO 

Let  a  pivot  be  generated  by 
revolving  the  curve  around  its 
axis  SL.  As  in  the  case  of  the 
conical  pivot  it  can  be  proved 
that  the  normal  pressure  on 
an  element  of  convex  surface  is 
SWrdr 


Fig.  49. 


dP  = 


.(a) 


Let  the  normal  wear  of  the  pivot  be  assumed  to  be 
proportional  to  this  normal  pressure  and  to  the  velocity 
of  the  rubbing  surfaces,  i.  e.  normal  wear  proportional 
to  pr,  then  is  the  vertical  wear  proportional  to 


pr 
sinff 


But 


sinO 


is  a  constant,  therefore  the  vertical 


wear  will  be  the  same  at  all  points.  This  is  the 
characteristic  feature  and  advantage  of  this  form  of 
pivot. 

As  shown  in  equation  (a) 
dp     SWr.dr 

=~~ 


SWfr.rdr 


and 


d\  — 


.(72) 


T  is  thus  shown  to  be  independent  of  d9  or  of  the 
length  of  pivot  used. 


116  MACHINE  DESIGN. 

This  pivot  is  sometimes  wrongly  called  antifriction. 
As  will  be  seen  by  comparing  equations  (68)  and  (72) 
the  moment  of  friction  is  fifty  per  cent,  greater  than 
that  of  the  common  flat  pivot. 

The  distinct  advantage  of  the  Schiele  pivot  is  in  the 
fact  that  it  maintains  its  shape  as  it  wears  and  is  self- 
adjusting.  It  is  an  expensive  bearing  to  manufacture 
and  is  seldom  used  on  that  account. 

It  is  not  suitable  for  a  bearing  where  most  of  the 
pressure  is  side  ways. 

56.  Multiple  Bearings.  To  guard  against  abrasion 
in  flat  pivots  a  series  of  rubbing  surfaces  which  divide 
the  wear  is  sometimes  provided.  Several  flat  discs 
placed  beneath  the  pivot  and  turning  indifferently, 
may  be  used.  Sometimes  the  discs  are  made  alter- 
nately of  a  hard  and  a  soft  material.  Bronze,  steel 
and  raw  hide  are  the  more  common  materials. 

Notice  in  this  connection  the  button  or  washer  at 
the  outer  end  of  the  head  spindle  of  an  engine  lathe 
and  the  loose  collar  on  the  main  journal  of  a  milling 
machine.  See  Figs.  41  and  42.  Pivots  are  usually 
lubricated  through  a  hole  at  the  center  of  the  bearing 
and  it  is  desirable  to  have  a  pressure  head  on  the  oil  to 
force  it  in. 

The  compound  thrust  bearing  generally  used  for 
propeller  shafts  consists  of  a  number  of  collars  of  the 
same  size  forged  on  the  shafts  at  regular  intervals  and 
dividing  the  end  thrust  between  them,  thus  reducing 
the  intensity  of  pressure  to  a  safe  limit  without  making 
the  collars  unreasonably  large. 

A  safe  value  for  p  the  intensity  of  pressure  is,  ac- 
cording to  Whitham,  60  Ib.  per  sq.  in.  for  high  speed 
engines. 


MULTIPLE  BEARINGS. 

A  table  given  by  Prof.  Jones  in  his  book  on  Machine 
Design  shows  the  practice  at  the  Newport  News  ship- 
yards 011  marine  engines  of  from  250  to  5000  H.  P. 
The  outer  diameter  of  collars  is  about  one  and  one-half 
times  the  diameter  of  the  shafts  in  each  case  and  the 
number  of  collars  used  varies  from  6  in  the  smallest 
engine  to  11  in  the  largest.  The  pressure  per  sq.  in.  of 
bearing  surface  varies  from  18  to  46  Ib.  with  an 
average  value  of  about  32  Ib. 

The  hydraulic  foot  step  sometimes  used  for  the 
vertical  shafts  of  turbines  is  in  effect  a  rotating  plunger 
supported  by  water  pressure  underneath  and  so  packed 
in  its  bearing  as  to  allow  a  slight  leakage  of  water  for 
cooling  and  lubricating  the  bearing  surfaces. 

PROBLEMS. 

1.  Design  and  draw  to  full  size  a  Schiele  pivot  for  a  water 
wheel  shaft  4  inches  in  diameter,  the  total  length  of  the  bear- 
ing being  3  inches. 

Calculate  the  horse-power  expended  in  friction  if  the  total 
vertical  pressure  on  the  pivot  is  two  tons  and  the  wheel  makes 
150  revs,  per  min.  and  assuming  /=. 25  for  metal  on  wet  wood. 

2.  Compare  the  friction  of  the  pivot  in  Prob.  1,  with  that  of 
a  flat  collar  of  the  same  projected  area  and  also  with  that 
of  a  conical  pivot  having  «=30  deg. 

3.  Design  a  compound  thrust  bearing  for  a  propeller  shaft 
the  diameters  being  14  and  21  inches,  the  total  thrust  being 
80,000  Ibs.  and  the  pressure  40  Ib.  per  sq.  in. 

Calculate  the  horse-power  consumed  in  friction  and  compare 
with  that  developed  if  a  single  collar  of  same  area  had  been 
used.  Assume  /=. 05  and  rev.  per  min.  =120. 


CHAPTER  VIII. 

BALL  AND  ROLLER  BEARINGS. 

57.  General  Principles.     The  object  of  interposing  a 
ball  or  roller  between  a  journal  and  its  bearing,  is  to 
substitute  rolling  for  sliding  friction  and  thus  to  re- 
duce the  resistance.     This  can  be  done  only  partially 
and  by  the  observance  of  certain  principles.     In  the 
first  place  it  must  be  remembered  that  each  ball  can 
roll  about  but  one  axis  at  a  time  ;  that  axis  must  be 
determined  and  the  points  of  contact  located  accord- 
ingly. 

Secondly,  the  pressure  should  be  approximately 
normal  to  the  surfaces  at  the  points  of  contact 

Finally  it  must  be  understood,  that  on  account  of 
the  contact  surfaces  being  so  minute,  a  comparatively 
slight  pressure  will  cause  distortion  of  the  balls  and  an 
entire  change  in  the  conditions. 

58.  Journal  Bearings.     These  may  be  either  two, 
three  or  four  point,  so  named  from  the  number  of 
points  of  contact  of  each  ball. 

The  axis  of  the  ball  may  be  assumed  as  parallel  or 
inclined  to  the  axis  of  the  journal  and  the  points  of 
contact  arranged  accordingly.  The  simplest  form  con- 
sists of  a  plain  cylindrical  journal  running  in  a  bearing 
of  the  same  shape  and  having  rings  of  balls  interposed. 
The  successive  rings  of  balls  should  be  separated  \)y 
thin  loose  collars  to  keep  them  in  place.  These  collars 
are  a  source  of  rubbing  friction,  and  to  do  away  with 
them  the  balls  are  sometimes  run  in  grooves  either  in 
journal,  bearing  or  both. 
118 


JOURNAL  BEARINGS. 


119 


Fig.  50  shows  a  bearing  of  this  type,  there  being 
three  points  of  contact  and  the  axis  of  ball  being 
parallel  to  that  of  journal. 

The  bearings  so  far 
mentioned    have    no 
„   means  of  adjustment 
y  for     wear.       Conical 
-J   bearings,  or  those  in 
which  the  axes  of  the 
balls  meet  in  a  com- 
mon    point,     supply 


Fig.  50. 


this  deficiency.  In  designing  this  class  of  bearings, 
either  for  side  or  end  thrust,  the  inclination  of  the 
axis  is  assumed  according  to  the  obliquity  desired  and 
the  points  of  contact  are  then  so  located  that  there 
shall  be  no  slipping. 

Tig.  51  illustrates  a  common  form  of  adjustable  or 
cone  bearing  and  shows  the  method  of  designing  a 
three  point  contact.  A  C 
is  the  axis  of  the  cone, 
while  the  shaded  area  is  a 
section  of  the  cup,  so 
called.  Let  a  and  b  be 
two  points  of  contact  be- 
tween ball  and  cup.  Draw 
the  line  a  b  and  produce 
to  cut  axis  in  A.  Through 
the  center  of  ball  draw 
the  line  A  B ;  then  will 


f       4/^m 

(T 


u-c 


Fig.  51. 


this  be  the  axis  of  rotation  of  the  ball  and  a  c,  b  d 
will  be  the  projections  of  two  circles  of  rotation.  As 
the  radii  of  these  circles  have  the  same  ratio  as  the  radii 
of  revolution  an,  b  m,  there  will  be  no  slipping  and 
the  ball  will  roll  as  a  cone  inside  another  cone.  The 


120 


MACHINE  DESIGN. 


exact  location  of  the  third  point  of  contact  is  not 
material.  If  it  were  at  c,  too  much  pressure  would 
come  on  the  cup  at  b  ;  if  at  d  there  would  be  an  excess 
of  pressure  at  a,  but  the  rolling  would  be  correct  in 
either  case.  A  convenient  method  is  to  locate  p  by 
drawing  A  D  tangent  to  ball  circle  as  shown.  It  is 
recommended  however  that  the  two  opposing  surfaces 
at  p  and  b  or  a  should  make  with  each  other  an  angle 
of  not  less  than  25  deg.  to  avoid  sticking  of  the  ball. 

To  convert  the  bearing  just  shown  to  four  point 
contact,  it  would  only  be  necessary  to  change  the  one 
cone  into  two  cones  tangent  to  the  ball  at  c  and  d. 

To  reduce  it  to  two  point  contact  the  points  a  and  b 
are  brought  together  to  a  point  opposite  p.  As  in  this 
last  case  the  ball  would  not  be  confined  to  a  definite 
path  it  is  customary  to  make  one  or  both  surfaces  con- 
cave conoids  with  a  radius  about  three  fourths  the 
diameter  of  the  ball.  See  Fig.  52. 

59.  Step-Bearings.  The 
same  principles  apply  as 
in  the  preceding  article 
and  the  axis  and  points 
of  contact  may  be  varied 
in  the  same  way.  The 
most  common  form  of 
step-bearing  consists  of 
two  flat  circular  plates 
Fig.  52.  separated  by  one  or  more 

rings  of  balls.  Each  ring  must  be  kept  in  place  by 
one  or  more  loose  retaining  collars,  and  these  in  turn 
are  the  cause,  of  some  sliding  friction.  This  is  a  bear- 
ing with  two  point  contact  and  the  balls  turning  on 
horizontal  axes.  If  the  space  between  the  plates  is 
filled  with  loose  balls,  as  is  sometimes  done,  the  rubbing 


B 


STEP-BEARINGS.  121 

of  the  balls  against  each  other  will  cause  considerable 
friction. 

To  guide  the  balls  without  rubbing  friction  three 
point  contact  is  generally  used. 

Fig.  58  illustrates  a  bearing  of  this  character.     The 


Fig.  53. 

method  of  design  is  shown  in  the  figure,  the  principle 
being  the  same  as  in  Fig.  51.  By  comparing  the  letter- 
ing of  the  two  figures  the  similarity  will  be  readily  seen. 

This  last  bearing  may 
be  converted  to  four  point 
contact  by  making  the  upper 
collar  of  the  same  shape  as 
the  lower.  To  guide  the 
balls  in  two  point  contact 
use  is  sometimes  made  of 
a  cage  ring,  a  flat  collar 
drilled  with  holes  just  a 
trifle  larger  than  the  balls 
and  disposing  them  either 
in  spirals  or  in  irregular 
order.  See  Fig.  54. 

This  method  has   the  advantage    of  making  each 


Fig.  54. 


MACHINE  DESIGN. 

ball  move  in  a  path  of  different  radius  thus  securing 
more  even  wear  for  the  plates. 

60.  Materials  and  Wear.  The  balls  themselves  are 
always  made  of  steel,  hardened  in  oil,  tempered  and 
ground.  They  are  usually  accurate  to  within  one  ten 
thousandth  of  an  inch.  The  plates,  rings  and  journals 
must  be  hardened  and  ground  in  the  same  way  and 
perhaps  are  more  likely  to  wear  out  or  fail  than  the 
balls.  A  long  series  of  experiments  made  at  the  Case 
School  of  Applied  Science  on  the  friction  and  endur- 
ance of  ball  step-bearings  showed  some  interesting 
peculiarities. 

Using  flat  plates  with  one  circle  of  quarter  inch  balls 
it  was  found  that  the  balls  pressed  outward  on  the 
retaining  ring  with  such  force  as  to  cut  and  indent  it 
seriously.  This  was  probably  due  to  the  fact  that  the 
pressure  slightly  distorted  the  balls  and  changed  each 
sphere  into  a  partial  cylinder  at  the  touching  points. 
While  of  this  shape  it  would  tend  to  roll  in  a  straight 
line  or  a  tangent  to  the  circle.  Grinding  the  plates 
slightly  convex  at  an  angle  of  one  to  one  and-a-half 
degrees  obviated  the  difficulty  to  a  certain  extent. 
Under  even  moderately  heavy  loads  the  continued 
rolling  of  the  ring  of  balls  in  one  path  soon  damaged 
the  plates  to  such  an  extent  as  to  ruin  the  bearing. 

A  flat  bearing  filled  with  loose  balls  developed  three 
or  four  times  the  friction  of  the  single  ring  and  a  three 
point  bearing  similar  to  that  in  Fig.  53  showed  more 
than  twice  the  friction  of  the  two  point. 

A  flat  ring  cage  such  as  has  already  been  described 
was  the  most  satisfactory  as  regards  friction  and  en- 
durance. 

The  general  conclusions  derived  from  the  experi- 


ROLLER  BEARINGS.  123 

ments  were  that  under  comparatively  light  pressures 
the  balls  are  distorted  sufficiently  to  seriously  disturb 
the  manner  of  rolling  and  that  it  is  the  elasticity  and 
not  the  compressive  strength  of  the  balls  which  must 
be  considered  in  designing  bearings. 

61.  Design    of  Bearings.     Figures    on    the    direct 
crushing  strength  of  steel  balls  have  little  value  for  the 
designer.     For  instance  it  has  been  proved  by  numer- 
ous tests  that  the  average  crushing  strengths  of  J  inch 
and  |  inch   balls   are   about   Y500   Ib.  and  15000   Ib. 
respectively.     Experiments  made  by  the  writer  show 
that  a  \  inch  ball  loses  all  value  as  a  transmission 
element  on  account  of  distortion,  at  any  load  of  more 
than  100  Ib. 

Prof.  Gray  states,  as  a  conclusion  from  some  ex- 
periments made  by  him,  that  not  more  than  40  Ib.  per 
ball  should  be  allowed  for  f  inch  balls. 

This  distortion  doubtless  accounts  for  the  failure  of 
theoretically  correct  bearings  to  behave  as  was  ex- 
pected of  them.  Ball  bearings  should  be  designed  as 
has  been  explained  in  the  preceding  articles  and  then 
only  used  for  light  loads. 

62.  Roller  Bearings.     The  principal    disadvanta^ 
of  ball  bearings  lies  in  the  fact  that  contact  is  only  , 
a  point  and  that  even  moderate  pressure  causes  exc< . 
sive  distortion  and  wear.     The  substitution  of  cylinders 
or  cones  for  the  balls  is   intended  to  overcome  this 
difficulty. 

The  simplest  form  of  roller  bearing  consists  of  a 
plain  cylindrical  journal  and  bearing  with  small  cylin- 
drical rollers  interposed  instead  of  balls.  There  are 
two  difficulties  here  to  be  overcome.  The  rollers  tend 


124 


MACHINE  DESIGN. 


JLLi 


Lc 


to  work  endways  and  rub  or  score  whatever  retains 
them.  They  also  tend  to  twist  around  and  become 

unevenly  worn  or  even 
bent  and  broken,  unless 
held  in  place  by  some  sort 
of  cage.  In  short  they 
will  not  work  properly 
Fjg>  55<  unless  guided  and  any 

form    of    guide    entails 

sliding  friction.  The  cage  generally  used  is  a  cylin- 
drical sleeve  having  longitudinal  slots  which  hold 
the  rollers  loosely  and  prevent  their  getting  out  of 
place  either  sideways  or  endways. 

The  use  of  balls  or  convex  washers  at  the  ends  of  the 
rollers  has  been  tried  with  some  degree  of  success. 
See  Fig.  55.  Large  rollers  have  been  turned  smaller 
at  the  ends  and  the  bearings  then  formed  allowed  to 
turn  in  holes  bored  in  revolving  collars.  These  collars 
must  be  so  fastened  or  geared  together  as  to  turn  in 
unison. 

63.  Grant  Roller  Bearing.    The  Grant  roller  is  conical 
and  forms  an  intermediate  between  the  ball  and  the 
cylindrical  roller  having  some 
of   the   advantages   of   each. 
The   principle    is    much    the 
same  as  in  the  adjustable  ball 
bearing,  Fig.  52,  rolling  cones 
being   substituted    for   balls, 
Fig.  56.     The  inner  cone  turns 
loose    on    the    spindle.     The 
conical  rollers  are  held  in  posi- 
tion by  rings  at  each  end,  while  the  outer  or  hollow 
cone  ring  is  adjustable  along  the  axis. 


Fig.  56. 


HYATT  ROLLERS.  125 

Two  sets  of  cones  are  used  on  a  bearing,  one  at  each 
end  to  neutralize  the  end  thrust,  the  same  as  with  ball 
bearings. 

64.  Hyatt  Rollers.  The  tendency  of  the  rollers  to 
get  out  of  alignment  has  been  already  noticed.  The 
Hyatt  roller  is  intended  by  its  flexibility  to  secure 
uniform  pressure  and  wear  under  such  conditions.  It 
consists  of  a  flat  strip  of  steel  wound  spirally  about 
a  mandrel  so  as  to  form  a  continuous  hollow  cylinder. 
It  is  true  inform  and  comparatively  rigid  against  com- 
pression, but  possesses  sufficient  flexibility  to  adapt 
itself  to  slight  changes  of  bearing  surface. 

Experiments  made  by  the  Franklin  Institute  show 
that  the  Hyatt  roller  possesses  a  great  advantage  in 
efficiency  over  the  solid  roller. 

Testing  f  inch  rollers  between  flat  plates  under  loads 
increasing  to  550  Ib.  per  linear  inch  of  roller  developed 
co-efficients  of  friction  for  the  Hyatt  roller  from  23  to 
51  per  cent,  less  than  for  the  solid  roller.  Subsequent 
examination  of  the  plates  showed  also  a  much  more 
even  distribution  of  pressure  for  the  former. 

A  series  of  tests  were  conducted  by  the  writer  in 
1904-05  to  determine  the  relative  efficiency  of  roller 
bearings,  as  compared  with  plain  cast  iron  and  bab- 
bitted bearings  under  similar  conditions.1  The  bear- 
ings tested  had  diameters  of  1||,  2T\,  2T7-g-,  and  2{| 
inches  and  lengths  approximately  four  times  the 
diameters.  In  the  first  set  of  experiments  Hyatt  roller 
bearings  were  compared  with  plain  cast  iron  sleeves, 
at  a  uniform  speed  of  480  rev.  per  min.  and  under  loads 
varying  from  64  to  264  pounds.  The  cast  iron  bearings 
were  copiously  oiled. 

1  Machinery,  N.  Y.,  Oct.  1905. 


126 


MACHINE  DESIGN. 


As  the  load  was  gradually  increased,  the  value  of  / 
the  coefficient  of  friction  remained  nearly  constant 
with  the  plain  bearings,  but  gradually  decreased  in  the 
case  of  the  roller  bearings.  Table  XXIII.  gives  a 
summary  of  this  series  of  tests. 


TABLE  XXIII. 

COEFFICIENTS    OF  FRICTION    FOR    ROLLER    AND    PLAIN 
BEARINGS. 


Diameter 

of 
Journal. 

Hyatt  Bearing 

Plain  Bearing. 

Max. 

Min. 

Ave. 

Max. 

Min. 

Ave. 

1H 

.036 

.019 

.026 

.160 

.099 

.117 

2& 

.052 

.034 

.040 

.129 

.071 

.094 

2TV 

.041 

.025 

.030 

.143 

.076 

.104 

2}| 

.053 

.049 

.051 

.138 

.091 

.104 

The  relatively  high  value  of  /  in  the  2T3g-  and  2 
roller  bearings  were  due  to  the  snugness  of  the  fit  be- 
tween the  journal  and  the  bearing,  and  show  the  ad- 
visability of  an  easy  fit  as  in  ordinary  bearings. 

The  same  Hyatt  bearings  were  used  in  the  second 
set  of  experiments,  but  were  compared  with  the 
McKeel  solid  roller  bearings  and  with  plain  babbited 
bearings  freely  oiled.  The  McKeel  bearings  contained 
rolls  turned  from  solid  steel  and  guided  by  spherical 
ends  fitting  recesses  in  cage  rings  at  each  end.  The 
cage  rings  were  joined  to  each  other  by  steel  rods  par- 
allel to  the  rolls.  The  journals  were  run  at  a  speed  of 
560  rev.  per  min.  and  under  loads  varying  from  113  to 
456  pounds.  Table  XXIV.  gives  a  summary  of  the 
second  series  of  tests. 


ROLLER  STEP-BEARINGS. 


127 


TABLE  XXIV. 

COEFFICIENTS  OF  FRICTION  FOR  ROLLER,  AND  PLAIN 
BEARINGS. 


Diam. 
of 
Jo'rnal  . 

Hyatt  Bearing. 

McKeel  Bearing. 

Babbitt  bearing. 

Max. 

Min. 

Ave. 

Max. 

Min. 

Ave. 

Max. 

Min. 

Ave. 

1A 

2& 
2i7i 

.032 
.019 
.042 

.012 
.011 
.025 

.018 
.014 
.032 

.033 

.017 

.022 

.074 
.088 
.114 

.029 

.078 
.083 

.043 
.082 
.096 

.028 

.015 

.021 

2*i 

.029 

.022 

.025 

.039 

.019 

.027 

.125 

.089 

.107 

The  variation  in  the  values  for  the  babbited  bearing 
is  due  to  the  changes  in  the  quantity  and  temperature 
of  the  oil.  For  heavy,  pressures  it  is  probable  that  the 
plain  bearing  might  be  more  serviceable  than  the  others. 
Notice  the  low  values  for /in  Table  XXII. 

Under  a  load  of  470  pounds  the  Hyatt  bearing  de- 
veloped an  end  thrust  of  13.5  pounds  and  the  McKeel 
one  of  11  pounds. 

This  is  due  to  a  slight  skewing  of  the  rolls  and 
varies,  sometimes  reversing  in  direction. 

If  roller  bearings  are  properly  adjusted  and  not 
overloaded  a  saving  of  from  two-thirds  to  three-fourths 
of  the  friction  may  be  reasonably  expected. 

65.  Roller  Step-Bearings/  In  article  60  attention 
was  called  to  the  fact  that  the  balls  in  a  step-bearing 
under  moderately  heavy  pressures  tend  to  become 
cylinders  or  cones  and  to  roll  accordingly.  This  has 
suggested  the  use  of  small  cones  in  place  of  the  balls, 
rolling  between  plates  one  or  both  of  which  are  also  con- 
ical. A  successful  bearing  of  this  kind  with  short 


128 


MACHINE  DESIGN. 


cylinders  in  place  of  cones  is  used  by  the  Sprague-Pratt 
Elevator  Co.,  and  is  described  in  the  American  Machin- 
ist for  June  27,  1901.  The  rollers  are  arranged  in  two 
spiral  rows  so  as  to  distribute  the  wear  evenly  over  the 
plates  and  are  held  loosely  in  a  flat  ring  cage.  This 
bearing  has  run  well  in  practice  under  loads  double 
those  allowable  for  ball  bearings,  or  over  100  Ib.  per 
roll  for  rolls  one-half  inch  in  diameter  and  one-quarter 
inch  long. 

Fig.  57  illustrates  a  bearing  of  this  character.     Col- 
lars similar  to  this  have  been  used  in  thrust  bearings 

for  propeller  shafts.  The 
discussion  referred  to  in 
Art.  46  also  included  ball 
and  roller  bearings  and 
should  be  read  by  the  de- 
signer. Mr.  Mossberg,  de- 
signer of  the  roller  bear- 
ings of  that  name,  recom- 
mends rollers  of  spring 
tempered  tool  steel,  cages 
of  tough  bronze  and  boxes 
of  high  carbon  steel  with  a 
hard  temper.  Mr.  Charles 
R.  Pratt  reports  the  limit  of  work  for  ^  inch  balls  in 
thrust  bearings  to  be  100  pounds  per  ball  at  700  revolu- 
tions per  minute  and  6  inches  diameter  circle  of  rotation. 
Mr.  W.  S.  Eogers  gives  the  maximum  load  for  a 
1  inch  ball  as  1000  pounds  and  for  a  £  inch  ball  as  200 
pounds.  Mr.  Henry  Hess  states  that  in  a  roller  bear- 
ing one  fifth  of  the  number  of  rollers  multiplied  by  the 
length  and  diameter  of  one  roller  may  be  considered  as 
the  projected  area  of  the  journal.  For  ball  bearings 
one  fifth  the  total  number  of  balls  multiplied  by  the 


F< 


ROLLER  STEP-BEARINGS.  129 

square  of  the  ball  diameter  may  be  used  in  the  same 
way. 

Space  forbids  reference  to  all  of  the  many  varieties 
of  ball  and  roller  bearings  shown  in  manufacturers' 
catalogues.  These  are  all  subject  to  the  laws  and  limit- 
ations mentioned  in  this  chapter, 

While  such  bearings  will  be  used  more  and  more  in 
the  future,  it  must  be  understood  that  extremely  high 
speeds  or  heavy  pressures  are  unfavorable  and  in  most 
cases  prohibitive. 

Furthermore,  unless  a  bearing  of  this  character  is 
carefully  designed  and  well  constructed  it  will  prove 
to  be  worse  than  useless. 


CHAPTER  IX. 

SHAFTING,  COUPLINGS  AND  HANGERS. 

66.     Strength  of  Shafting. 

Let       D= diameter  of  the  driving  pulley  or  gear. 
N=  number  rev,  per  minute. 
P= force  applied  at  rim. 
T=  twisting  moment* 

The  distance  through  which  P  acts  in  one  minute  is 
irDN  inches  and  wor]s.= PirDN  in.  Ib.  per  min. 

PD 

But     — —  =T  the   moment,    and   2-n-N  =  the  angular 

velocity. 

.-.  work  =  moment  X  angular  velocity. 
One  horse  power =  33000  ft.  Ib.  per  min. 
=  396000  in.  Ib.  per  min. 


396000  ~  396000 

HP=^ ^ 

il*r  T     63Q25  HP 

~N~          

p    126050  HP  ,^-N 

D^ 

The  general  formula  for  a  circular  shaft  exposed  to 
torsion  alone  is 


130 


STRENGTH  OF  SHAFTING. 

But  r=63025 

where  N=no.  rev.  per  min. 
Substituting  in  formula  for  d 


^3210001    -  nearly  .    .     .     (76) 

S  may  be  given  the  following  values  : 

45000  for  common  turned  shafting. 
50000  for  cold  rolled  iron  or  soft  steel. 
65000  for  machinery  steel. 

It  is  customary  to  use  factors  of  safety  for  shafting 
as  follows  : 

Headshafts  or  prime  movers     15 
Line  shafting  10 

Short  counters  6 

The  large  factor  of  safety  for  head  shafts  is  used  not 
only  on  account  of  the  severe  service  to  which  such 
shafts  are  exposed,  but  also  on  account  of  the  incon- 
venience and  expense  attendant  on  failure  of  so  im- 
portant a  part  of  the  machinery.  The  factor  of  safety 
for  line  shafting  is  supposed  to  be  large  enough  to 
allow  for  the  transverse  stresses  produced  by  weight 
of  pulleys,  pull  of  belts,  etc.,  since  it  is  impracticable 
to  calculate  these  accurately  in  most  cases. 

Substituting  the  values  of  S  and  introducing  factors 
of  safety,  we  have  the  following  formulas  for  the  safe 
diameters  of  the  various  kinds  of  shafts. 


132 


MACHINE  DESIGN. 


TABLE  XXV. 
DIAMETERS   OF   SHAFTING. 


KIND  OF 
SHAFT. 


Head  Shaft. 


Line  Shaft. 


Counter  Shaft. 


MATERIAL. 


Com'n  Iron 


A  Y 


415 

5 


3.50 


HP 


Soft  Steel    Mach'y  Steel 


4.00 


3.38 


3.10 


The  Allis-Chalmers  Co.  base  their  tables  for  the 
horse  power  of  wrought  iron  or  mild  steel  shafting  on 
the  formula  HP=cd*N  where  c  has  the  following 
values : 

c 

Heavy  or  main  shafting         .008 
Shaft  carrying  gears  .010 

Light  shafting  with  pulleys  .013 
This  is  equivalent  to  using  values  of  S  as  2570  Ib. , 
3200  Ib.  and  41TO  Ib.  per  sq.  in.  in  the  respective  classes 
— and  would  give  for  co-efficients  in  Table  XXV.  the 
numbers  5,  4.64  and  4.25  which  are  somewhat  larger 
than  those  given  for  similar  cases  in  the  table. 

A  table  published  by  Wm.  Sellers  &  Co.  in  their 
shafting  catalogue — gives  the  horse  powers  of  iron  and 
steel  shafts  for  given  diameters  and  speeds.  An  invest!- 


COUPLINGS. 


133 


gation  of  the  table  shows  it  to  be  based  upon  a  value 
of  about  4000  Ib.  for  S  or  a  co-efficient  of  4.31  in  Table 
XXV. 

In  case  there  is  a  known  bending  moment  M,  com- 
bined with  a  known  twisting  moment  T,  then  a  re- 
sultant twisting  moment 


is  to  be  substituted  for  Tin  the  formulas  (73)  to  (75). 

Mr.  J.  B.  Francis  has  published  a  table  in  the  Journal 
of  the  Franklin  Institute  which  gives  the  greatest 
admissible  distance  between  bearings  for  line  shafts  of 
different  diameters,  when  subject  to  no  transverse 
forces  except  from  their  own  weight.  This  distance 
varies  from  16  feet  for  2  inch  shafts  up  to  26  feet  for 
9  inch  shafts,  the  span  being  proportional  to  the  cube 
root  of  the  diameter.  The  distance  should  be  much 
less  when  the  shaft  carries  numerous  pulleys  with 
their  belts. 

67.  Couplings.  The  flange  or  plate  coupling  is  most 
commonly  used  for  fastening  together  adjacent  lengths 
of  shafting. 

Fig.  58  shows  the  pro- 
portions of  such  a  coup- 
ling.    The    flanges    are 
turned  accurately  on  all    r- 
sides,   are  keyed  to  the   Q. 
shafts  and  the  two  are 
centered  by  the  projec- 
tion of  the   shaft  from 
one  part  into  the  other  Fig  58 

as    shown    at    A.     The 

bolts  are  turned  to  fit  the  holes  loosely  so  as  not  to  in- 
terfere with  the  alignment, 


134 


MACHINE  DESIGN. 


The  projecting  rim  as  at  B  prevents  danger  from 
belts  catching  on  the  heads  and  nuts  of  the  holts. 

The  faces  of  this  coupling  should  he  trued  up  in  a 
lathe  after  being  keyed  to  the  shaft. 

Jones  and  Laughlins  in  their  shafting  catalogue  give 
the  following  proportions  for  flange  couplings. 


Diam.  of  Shaft. 

Diani.  of  Hub. 

Length  of  Hub. 

Diam.  of 
Coupling. 

2 

4i 

3 

8 

21 

5| 

4 

10 

3 

6* 

5, 

12 

3* 

8 

6i 

14 

4 

9 

7 

16 

5 

HI 

8f 

20 

There  are  five  bolts  in  each  coupling. 
The  sleeve  coupling  is  neater  in  appearance  than  the 
flange  coupling  but  is  more  complicated  and  expensive. 
In  Fig.  59  is  illustrated  a  neat  and  effective  coupling 


B 


Fig.  59. 


of  this  type.  It  consists  of  the  sleeve  S  bored  with 
two  tapers  and  two  threaded  ends  as  shown.  The  two 
conical,  split  bushings  BB  are  prevented  from  turning 
by  the  feather  key  K  and  are  forced  into  the  conical 
recesses  by  the  two  threaded  collars  C  C  and  thereby 


CLUTCHES.  135 

clamped  firmly  to  the  shaft.  The  key  K  also  nicks 
slightly  the  center  of  the  main  sleeve  S,  thus  locking 
the  whole  combination. 

Couplings  similar  to  this  have  been  in  use  in  the 
Union  Steel  Screw  Works,  Cleveland,  Ohio,  for  many 
years  and  have  given  good  satisfaction. 

The  Sellers  coupling  is  of  the  type  illustrated  in  Fig. 
59,  but  is  tightened  by  three  bolts  running  parallel  to 
the  shaft  and  taking  the  place  of  the  collars  C  C. 

In  another  form  of  sleeve 
coupling  the  sleeve  is  split 
and  clamped  to  the  shaft  by 
bolts  passing  through  the 
two  halves  as  illustrated  in 
Fig.  60. 

The  "muff"  coupling,  as     [  i I    L J 

its  name  implies  is  a  plain  pjg> 

sleeve  slipped  over  the  shafts 

at  the  point  of  junction,  accurately  fitted  and  held  by 

a  key  running  from  end  to  end.     It  may  be  regarded 

as  a  permanent  coupling  since  it  is  not  readily  removed. 

68.  Clutches.  By  the  term  clutch,  is  meant  a  coup- 
ling which  may  be  readily  disengaged  so  as  to  stop  the 
following  shaft  or  pulley.  Clutch  couplings  are  of 
two  kinds,  positive  or  jaw  clutches  and  friction 
clutches. 

The  jaw  clutch  consists  of  two  hubs  having  sector 
shaped  projections  on  the  adjacent  faces  which  may 
interlock.  One  of  the  couplings  can  be  slid  on  its  shaft 
to  and  from  the  other  by  means  of  a  loose  collar  and 
yoke,  so  as  to  engage  or  disengage  with  its  mate. 
This  clutch  has  the  serious  disadvantage  of  not  being 
readily  engaged  when  either  shaft  is  in  motion. 


136 


MACHINE  DESIGN. 


Friction  clutches  are  not  so  positive  in  action  but  can 

be  engaged  without  difficulty  and  without  stopping 

the  driver. 

Three  different  classes  of  friction  clutches  may  be 

distinguished  according  as  the  engaging  members  are 

flat  rings,  cones  or  cylinders. 

The  Weston  clutch, 
Fig.  61,  belongs  to  the 
first  named  class.  A 
series  of  rings  inside  a 
>  sleeve  on  the  follower 
B  interlocks  with  a 
similar  series  outside  a 
smaller  sleeve  on  the 
driver  A  somewhat  as 


Fig.  61. 


in     a    thrust    bearing 
(Art.   56).     Each    ring 
can  slide  on  its  sleeve  but  must  rotate  with  it. 

When  the  parts  A  and  B  are  forced  together  the 
rings  close  up  and  engage  by  pairs,  producing  a 
considerable  turning  moment  with  a  moderate  end 
pressure.  Let : 

P= pressure  along  axis. 
n= number  of  pairs  of  surfaces  in  contact. 
f  =  coefficient  of  friction. 
r=mean  radius  of  ring. 
T— turning  moment. 
Then  will  : 

T=Pfnr (77) 

If  the  rings  are  alternately  wood  and  iron,  as  is 
usually  the  case,  /  will  have  values  ranging  from  0.25 
to  0.50. 

The  cone  clutch  consists  of  two  conical  frustra,  one 
external  and  one  internal,  engaging  one  another  and 


CLUTCHES. 


137 


driving  by  friction.  Using  the  same  notation  as  before, 
and  letting  a=  angle  between  element  of  cone  and 
axis,  the  normal  pressure  between  the  two  surfaces 

JL  J       J.'K  „       £^'^2-l^^          »-,»411     "U^  -*-.! 


will  be  :  — ^ — •  and  the  friction  will  be  : 

sin  a.  sin  a 


Therefore  : 


T  = 


.(T8) 


sin  a 

a  should   slightly  exceed   5  deg.  to  prevent   sticking 
and/ will  be  at  least  0.10  for  dry  iron  on  iron. 

Substituting  /  =  0.10  and  sin  a  =0.125  we  have 
T  =  O.SPr  as  a  convenient  rule  in  designing. 

Fig.  62  illustrates  the  type  of  clutch  more  generally 
used  on  shafting  for 
transmitting     mod- 
erate  quantities    of 
power. 

As  shown  in  the 
figure  one  member 
is  attached  to  a  loose 
pulley  on  the  shaft, 
but  this  same  type 
can  be  used  for  con- 
necting two  in- 
dependent shafts. 

The  ring  or  hoop 
H,  finished  inside 
and  out,  is  gripped 
at  intervals  by  pairs 
of  jaws  JJ  having  wooden  faces. 

These  jaws  are  actuated  as  shown  by  toggles  and 
levers  connected  with  the  slip  ring  R.  The  toggles 
are  so  adjusted  as  to  pass  by  the  center  and  lock  in  the 
gripping  position. 

These  clutches  are    convenient    and    durable    but 


Fig.  62. 


138 


MACHINE  DESIGN. 


occupy  considerable  room  in  proportion  to  their  trans- 
mitting power.  The  Weston  clutch  is  preferable  for 
heavy  loads. 

The  roller  clutch  is  much  used  on  automatic  ma- 
chinery as  it  combines  the  advantages  of  positive 
driving  and  friction  engagement.  A  cylinder  on  the 
follower  is  embraced  by  a  rotating  ring  carried  by  the 
driver. 

The  ring  has  a  number  of  recesses  on  its  inner  surface 
which  hold  hardened  steel  rollers.  These  recesses  being 
deeper  at  one  end  allow  the  rollers  to  turn  freely  as 
long  as  they  remain  in  the  deep  portions. 

The  bottom  of  the  recess  is  inclined  to  the  tangent  of 
the  circle  at  an  angle  of  from  9  to  14  deg. 

When  by  suitable  mechanism  the  rollers  are  shifted 
to  the  shallow  portions  of  the  recesses  they  are  im- 
mediately gripped  between  the  ring  and  the  cylinder 
and  set  the  latter  in  motion. 

A  clutch  of  this  type  is  almost  instantaneous  in  its 
action  and  is  very  powerful,  being  limited  only  by  the 
strength  of  the  materials  of  which  it  is  composed. 

Several  small  rolls  of  different  materials  and  diam- 
eters were  tested  by  the  writer  in  1905  with  the 
following  results : 

Material.       Diameter.       Length.        Set  load.       Ultimate  load. 


Cast  Iron 


Soft  Steel 
Hard  Steel 


0.375 

0.75 

1.125 

0.4375 

0.4375 

0.4375 


1.5 
1.5 
1.5 
1.5 
1.5 
1.5 


5500 

6800 

7800 

8800 

11100 

35000 


12400 
19500 
29700 
20000 


69.  Coupling  Bolts.     The  bolts  used  in  the  ordinary 
flange  couplings  are  exposed  to  shearing,   and   their 


SHAFTING  KEYS.  139 

combined  shearing  moment  should  equal  the  twisting 
moment  on  the  shaft. 

Let  n—  number  of  bolts. 
dt  =  diameter  of  bolt. 
D—  diameter  of  bolt  circle. 

We  will  assume  that  the  bolt  has  the  same  shearing 
strength  as  the  shaft.     The  combined  shearing  strengll 
of  the  bolts  is  .IStedinS  and  their  moment  of  resistance 
to  shearing  is 


This  last  should  equal  the  torsion  moment  of  the 
shaft  or 


Solving  for  dl  and  assuming  D=3d  as  an  average 
value,  we  have  d^—  ,—  •  .................  (79) 

v    0  1  1/ 

In  practice  rather  larger  values  are  used  than  would 
be  given  by  the  formula. 

70.  Shafting  Keys.  The  moment  of  the  shearing 
stress  on  a  key  must  also  equal  the  twisting  moment 
of  the  shaft. 

Let    b=  breadth  of  a  key. 
Z=  length  of  key. 
h=  total  depth  of  key. 
S'  =  shearing  strength  of  key. 
The  moment  of  shearing  stress  on  key  is 


and  this  must  equal  -gy  Usually       &=j- 

For  shafts  of  machine   steel   S=Sf,  and   for  iron 
shafts  $=f$'  nearly,  as  keys  should  always  be  of  steel. 


140  MACHINE  DESIGN. 

Substituting  these  values  and  reducing  : 

For  iron  shafting  l=l.2d  nearly. 

For  steel  shafting  1=1.  Qd  nearly,  as  the  least 

lengths  of  key  to  prevent  its  failing 
by  shear. 

If  the  key  way  is  to  be  designed 
for  uniform  strength,  the  shearing 
area  of  the  shaft  on  the  line  A  B, 
Fig.  63,  should  equal  the  shearing 
area  of  the  key,  if  shaft  and  key  are 
of  the  same  material  and  AB= 


Fig.  63. 

These  proportions  will  make  the 

depth  of  key  way  in  shaft  about  =|6  and  would  be 
appropriate  for  a  square  key. 

To  avoid  such  a  depth  of  key  way  which  might 
weaken  the  shaft,  it  is  better  to  use  keys  longer  than 
required  by  preceding  formulas.  In  American  practice 
the  total  depth  of  key  rarely  exceeds  f  &  and  one-half 
of  this  depth  is  in  shaft. 

To  prevent  crushing  of  the  key  the  moment  of  the 
compressive  strength  of  half  the  depth  of  key  must 
equal  T. 

dlh^  0     Sd3  f  . 

or  2X-2X&=-0  ........  ,..-(a) 

where  Se  is  the  compressive  strength  of  the  key. 

For  iron  shafts  Se=2S 

g 
and  for  steel  shafts        $c=n$ 

a 

Substituting  values  of  Sc  and  assuming  h=^b=-f^d 
we  have 

Iron  shafts  l=2.5d  nearly. 

Steel  shafts  l=3\d  nearly,  as  the  least 

length  for  flat  keys  to  prevent  lateral  crushing. 


HANGERS  AND  BOXES. 


141 


The  above  refers  to  parallel  keys.  Taper  keys  have 
parallel  sides,  but  taper  slightly  between  top  and  bot- 
tom. When  driven  home  they  have  a  tendency  to  tip 
the  wheel  or  coupling  on  the  shaft.  This  may  be  par- 
tially obviated  by  using  two  keys  90  deg.  apart  so  as 
to  give  three  points  of  contact  between  hub  and  shaft. 
The  taper  of  the  keys  is  usually  about  \  inch  to  one 
foot. 

The  Woodruff  key  is  sometimes  used  on  shafting. 
As  may  be  seen  in  Fig.  64  this 
key  is   semi-circular   in   shape 
and  fits  a  recess  sunk  in  the 
shaft  by  a  milling  cutter. 


Fig.  64. 


71.  Hangers  and  Boxes.  Since 
shafting  is  usually  hung  to  the 
ceiling  and  walls  of  buildings 
it  is  necessary  to  provide  means 
for  adjusting  and  aligning  the 
bearings  as  the  movement  of 
the  building  disturbs  them. 
Furthermore  as  line  shafting 
is  continuous  and  is  not  per- 
fectly true  and  straight,  the 
bearings  should  be  to  a  certain 

extent  self-adjusting.  Eeliable  experiments  have 
shown  that  usually  one-half  of  the  power  developed  by 
an  engine  is  lost  in  the  friction  of  shafting  and  belts. 
It  is  important  that  this  loss  be  prevented  as  far  as 
possible. 

The  boxes  are  in  two  parts  and  may  be  of  bored  cast- 
iron  or  lined  with  Babbitt  metal.  They  are  usually 
about  four  diameters  of  the  shaft  in  length  and  are 
oiled  by  means  of  a  well  and  rings  or  wicks.  (See  Art. 


142 


MACHINE  DESIGN. 


44.)  The  best  method  of  supporting  the  box  in  the 
hanger  is  by  the  ball  and  socket  joint  ;  all  other  con- 
trivances such  as  set  screws  are  but  poor  substitutes. 

Fig.  65  shows  the 
usual  arrangement  of 
the  ball  and  socket. 

A  and  B  are  the  two 
parts  of  the  box.  The 
center  is  cast  in  the 
shape  of  a  partial 
sphere  with  C  as  a 
center  as  shown  by 
the  dotted  lines.  The 
two  sockets  S  S  can  be 
adjusted  vertically  in  Fig.  65. 

the  hanger  by  means 

of  screws  and  lock  nuts.  The  horizontal  adjustment 
of  the  hanger  is  usually  effected  by  moving  it  bodily 
on  the  support,  the  bolt  holes  being  slotted  for  this 
purpose. 

Counter  shafts  are  short  and  light  and  are  not  subject 

to  much  bending.  Con- 
sequently there  is  not  the 
same  need  of  adjustment 
as  in  line  shafting. 

In  Fig.  66  is  illustrated 
a  simple  bearing  for 
counters.  The  solid  cast 
iron  box  B  with  a  spheric- 
al center  is  fitted  directly 
in  a  socket  in  the  hanger 
Fig.  66.  H  and  held  in  position  by 

the  cap  C  and  a  set  screw. 
There  is  not  space  here  to  show  all  the  various  forms 


HANGERS  AND  BOXES. 


143 


of  hangers  and  floor  stands  arid  reference  is  made  to 
the  catalogues  of  manufacturers.  Hangers  should  be 
symmetrical,  i.  e.,  the  center  of  the  box  should  be  in  a 
vertical  line  with  center  of  base.  They  should  have 
relatively  broad  bases  and  should  have  the  metal  dis- 
posed to  secure  the  greatest  rigidity  possible.  Cored 
sections  are  to  be  preferred. 


Fig.  67. 


e 

Fig.  68. 


Fig.  67  illustrates  the  proportions  of  a  Sellers  line- 
shaft  hanger.  This  type  is  also  made  with  the  lower 
half  removable  so  as  to  facilitate  taking  down  the 
shaft. 

Fig.  68  shows  the  outlines  of  a  hanger  for  heavy 

shafting  as  manu- 
factured by  the  Jones 
&  Laughlins  Com- 
pany while  Fig.  69  il- 
lustrates the  design  of 
Fig.  69.  the  box  with  oil  wells 

and  rings. 
The  open  side  hanger  is  sometimes  adopted  on  ao 


MACHINE  DESIGN. 


count  of  the  ease  with  which  the  shaft  can  be  removed, 

but  it  is  much 
less  rigid  than  the 
closed  hanger  and 
is  suitable  only 
for  light  shafting. 
The  countershaft 
hanger  shown  in 
Fig.  70  is  simple, 
strong  and  sym- 
metrical and  is  a 
great  improve- 
ment over  those 
using  pointed  set 
screws  for  pivots. 

Hangers  similar  to  this  are  used  by  the   Brown   & 
Sharpe  Mfg.  Co.  with  some  of  their  machines. 

PROBLEMS. 

1.  Calculate  the  safe  diameters  of  head  shaft  and  three  line 
shafts  for  a  factory,  the  material  to  be  rolled  iron  and  the 
speeds  and  horse-powers  as  follows  : 


200  rev.  per  min. 
120  rev.  per  min. 
250  rev.  per  min. 
200  rev.  per  min. 

least   two    lines   of 


Head  shaft  100  H  P 

Machine  shop  30  H  P 

Pattern  shop  50  H  P 

Forge  shop  20  H  P 

2.  Determine   the     horse-power   of  at 
shafting  whose  speeds  and  diameters  are  known. 

3.  Design  and  sketch  to  scale  a  flange  coupling  for  a  three 
inch  line  shaft  including  bolts  and  keys. 

4.  Design  a  sleeve  coupling  for  the  foregoing,  different  in 
principle  from  the  ones  shown  in  the  text. 

5.  A   four-inch  steel  head  shaft    makes  100  rev.  per  min. 
Find  the  horse-power  which  it  will  safely  transmit,  and  design 
a  Weston  ring  clutch  capable  of  carrying  the  load, 


HANGERS  AND  BOXES.  145 

There  are  to  be  six  wooden  rings  and  five  iron  rings  of  12  in. 
mean  diameter.  Find  the  moment  carried  by  each  pair  of 
surfaces  in  contact  and  the  end  pressure  required. 

6.  Find  mean  diameter  of  a  single  cone  clutch  for  same 
shaft  with  same  end  pressure. 

7.  Find   radial   pressure   required   for  a   clutch   like    that 
shown  in  Fig.  62,  the  ring  being  24  in.  in  mean  diameter  and 
there  being  four  pairs  of  grips.     Other  conditions  as  in  pre- 
ceding problems. 

8.  Select  the  line  shaft  hanger  which  you  prefer  among 
those  in  the  laboratories  and  make  sketch  and  description  of 
the  same. 

9.  Do.  for  a  countershaft  hanger. 

10.  Explain  in  what  way  a  floor-stand  differs  from  a  hanger. 


CHAPTER  X. 

GEARS,  PULLEYS  AND  CRANKS. 

72.  Gear  Teeth.  The  teeth  of  gears  may  be  either 
cast  or  cut,  but  the  latter  method  prevails,  since  cut 
gears  are  more  accurate  and  run  more  smoothly  and 
quietly?  The  proportions  of  the  teeth  are  essentially 
the  same  for  the  two  classes,  save  that  more  back  lash 
must  be  allowed  for  the  cast  teeth.  The  circular  pitch 
is  obtained  by  dividing  the  circumference  of  the  pitch 
circle  by  the  number  of  teeth.  The  diametral  pitch  is 
obtained  by  dividing  the  number  of  teeth  by  the  diam- 
eter of  the  pitch  circle  and  equals  the  number  of  teeth 
per  inch  of  diameter.  The  reciprocal  of  the  diametral 
pitch  is  sometimes  called  the  module.  The  addendum 
is  the  radial  projection  of  the  tooth  beyond  the  pitch 
circle,  the  dedendum  the  corresponding  distance  inside 
the  pitch  circle.  The  clearance  is  the  difference  be- 
tween the  dedendum  and  addendum  ;  the  back  lash 
the  difference  between  the  widths  of  space  and  tooth 
on  the  pitch  circle. 

Let  circular  pitch  =  p. 

module  =^=m. 

7T 

diametral  pitch  =—  =  — 
p    m 

addendum  =a. 
dedendum  or  flank  =/. 
clearance  =f—a=c. 
height  =a+f=h. 

width  =w.  (See  Fig.  72.) 

146 


GEAR  TEETH. 
The  usual  rule  for  standard  cut  teeth  is  to  make 

w  =  &    allowing  110  calculable  back-lash,  to  make  a=m 

2i 

and   f=-^-  or  h=^m  and  clearance  =~- 

8  o 

There  is  however  a  marked  tendency  at  the  present 
time  towards  the  use  of  shorter  teeth.  The  reasons 
urged  for  their  adoption  are  :  first,  greater  strength 
and  less  obliquity  of  action  ;  second,  less  expense  in 
cutting.  *  Several  systems  have  been  proposed  in  which 
the  height  of  tooth  h  varies  from  0.425p  to  0.55p. 

According  to  the  latter  system  a=0.25p,  /=0.3p, 
and  c=.05p. 

In  modern  practice  the  diametral  pitch  is  a  whole 
number  or  a  common  fraction  and  is  used  in  describing 
the  gear.  For  instance  a  3  pitch  gear  is  one  having 
3  teeth  per  inch  of  diameter.  The  following  table 
gives  the  pitches  in  common  use  and  the  proportions 
of  long  and  short  teeth. 

If  the  gears  are  cut,  w=^  ;  if  cast  gears  are  used, 

2t 
to  0.48p. 


*  See  American  Machinist,  Jan.  7,  1897,  p.  6. 


148 


MACHINE  DESIGN. 


TABLE  XXVI. 

PROPORTIONS  OF  GEAR  TEETH. 


PITCH. 

STANDARD  TEETH. 

SHORT  TEETH. 

Diame- 
tral. 

Circular. 

Addend. 
a 

Height. 

Clear- 
ance. 
c 

Addend. 
a 

Height. 
n 

Clear- 
ance. 
c 

* 

6.283 

2. 

4.25 

0.25 

1.571 

3.456 

.  0.314 

f 

4.189 

1.33 

2.82 

0.167 

1.047 

2.303 

0.209 

1 

3.142 

1. 

2.125 

0.125 

0.785 

1.728 

0.157 

1± 

2.513 

0.8 

1.7 

0.1 

0.628 

1.383 

0.125 

H 

2.094 

0.667 

1.415 

0.083 

0.524 

1.152 

0.105 

if 

1.795 

0.571 

1.212 

0.071 

0.449 

0.988 

0.09 

2 

1.571 

0.5 

1-062 

0.062 

0.392 

0.863 

0.078 

8* 

1.396 

0  .  445 

0.945 

0.056 

0.349 

0.768 

0.070 

24 

1.257 

0.4 

0  85 

0.05 

0.314 

0.691 

0.063 

2f 

1.142 

0.364 

0.775 

0.045 

0.286 

0.629 

0.057 

3 

1.047 

0.333 

0.708 

0.042 

0.262 

0.576 

0.052 

3| 

0.898 

0.286 

0.608 

0.036 

0.224 

0.494 

0.045 

4 

0.785 

0  25 

0.531 

0.031 

0.196 

0.432 

0.039 

5 

0.628 

0.2 

0.425 

0.025 

0.157 

0.345 

0.031 

6 

0.524 

0.167 

0.354 

0.021 

0.131 

0.288 

0.026 

7 

0.449 

0.143 

0.304 

0.018 

0.112 

0.246 

0.022 

8 

0.393 

0.125 

0.266 

0.016 

0.098 

0.216 

0.020 

9 

0.349 

0.111 

0.236 

0.014 

0.087 

0.191 

0.017 

10 

0.314 

0.1 

0.212 

0.012 

0.079 

0.174 

0.016 

11 

0.286 

0.091 

0.193 

0.011 

0.071 

0.156 

0.014 

12 

0.262 

0.0834 

0.177 

0.010 

0.065 

0.143 

0.013 

13 

0.242 

0.077 

0.164 

0.010 

0.060 

0.132 

0.012 

14 

0.224 

0.0715 

0.152 

0.009 

0.056 

0.123 

0.011 

15 

0.209 

0.0667 

0.142 

0.008 

0.052 

0.114 

0.010 

16 

0.196 

0.0625 

0.133 

0.008 

0.049 

0.108 

0.010 

73.  Strength  of  Teeth. 

Let  P  =  total  driving  pressure  on  wheel  at  pitch  cir- 
cle. This  may  be  distributed  over  two  or  more  teeth, 
but  the  chances  are  against  an  even  distribution. 

Again,  in  designing  a  set  of  gears  the  contact  is 
likely  to  be  confined  to  one  pair  of  teeth  in  the  smaller 
pinions, 


STRENGTH  OF  TEETH. 


149 


Each  tooth  should  therefore  be  made  strong  enough 
to  sustain  the  whole  pressure. 

Rough  Teeth.  The  teeth  of  pattern  molded  gears  are 
apt  to  be  more  or  less  irregular  in  shape,  and  are 
especially  liable  to  be  thicker  at  one  end  on  account  of 
the  draft  of  the  pattern. 

In  this  case  the  entire  pressure  may  come  on  the 
outer  corner  of  a  tooth  and  tend  to  cause  a  diagonal 
fracture. 

Let  O  in  Fig.  71  be  the  point  of  application  of  the 
pressure  P,  and  AB  the  line  of  probable  fracture. 


Drop  the 

1   CD  on 
AB 

Let  AB  =  x 
and 


angle 
CAD  =  a 


t 


Fig.  71. 

The  bending  moment  at 
section  AB  is  M=Py,  and 
the  moment  of  resistance  is 


Fig.  72.  were    S  =  safe     transverse 

strength  of  material. 


and 


(a) 


150  MACHINE  DESIGN. 

If  P  and  w  are  constant,  then  S  is  a  maximum 

when      -2-  is  a  maximum. 
# 

But  11  =  h  sina  and  x=—  —  - 

COSa 


cosa  which  is  a 

X 

maximum  when  a  =  45°   and  -^-  = 

£t/ 

o  r> 


Substituting  this  value  in  (a)  we 


3  P 
But  in  this  case  w  =A7p  and  therefore  $=          2  . 

ip 
and  p=3.684\S     ......  < 

diametral  pitch,        ^='853\/p    ......   (81) 

Unless  machine  molded  teeth  are  very  carefully 
made,  it  may  be  necessary  to  apply  this  rule  to  them 
as  well. 

Cut  Gears.  With  careful  workmanship  machine 
molded  and  machine  cut  teeth  should  touch  along  the 
whole  breadth.  In  such  cases  we  may  assume  a  line 
of  contact  at  crest  of  tooth  and  a  maximum  bending 
moment. 

M=Ph 

The  moment  of  resistance  at  base  of  tooth  is 
Ml  =  ^Sbw2 

when  5  is  the  breadth  of  tooth. 

In  most  teeth  the  thickness  at  base  is  greater  than 
w,  but  in  radial  teeth  it  is  less.  Assuming  standard 
proportions  for  cut  gears  : 


LEWIS'  FORMULAS.  151 


h  = 
w  =  .p 

and  substituting  above  : 
.6T65  p 


.......      (82) 

For  short  teeth  having  h  =  .55p  formula  (82)  reduces 
to: 

P  =  .0758&Sp     .     .  '  .......     (83) 

The  above  formulas  are  general  whatever  the  ratio 
of  breadth  to  pitch.  The  general  practice  in  this  coun- 
try is  to  make 

b=  3p 

Substituting  this  value  of  b  in  (82)  and  (83)  and 
reducing  : 

Long  teeth  :  p  =  2.326 


Short  teeth  :p  =  2.098 


J|  .........  (85) 


The  corresponding  formulas  for  the  diametral  pitch 
are  : 

Long  teeth  :L=  1.35    ^p'    '    '    .....    (86) 

Short  teeth:—  =1.49  j£  ........    (8Y) 

m  \p 

74.  Lewis'  Formulas.  The  foregoing  formulas  can 
only  be  regarded  as  approximate,  since  the  strength 
of  gear  teeth  depends  upon  the  number  of  teeth  in  the 
wheel  ;  the  teeth  of  a  rack  are  broader  at  the  base  and 
consequently  stronger  than  those  of  a  pinion.  This 
is  more  particularly  true  of  epicycloidal  teeth.  Mr. 
Wilfred  Lewis  has  deduced  formulas  which  take  into 


152  MACHINE  DESIGN. 

account  this  variation.     For  cut  spur  gears  of  standard 
dimensions  the  Lewis  formula  is  as  follows  : 


.(88) 


where  n= number  of  teeth. 

This  formula  reduces  to  the  same  as  (82),  for  %=14 
nearly. 

Formula  (82)  would  then  properly  apply  only  to 
small  pinions,  but  as  it  would  err  on  the  safe  side  for 
larger  wheels,  it  can  be  used  where  great  accuracy  is 
not  needed.  The  same  criticism  applies  to  the  other 
formulas  in  Art.  73. 

The  value  of  S  used  should  depend  on  the  material 
and  on  the  speed. 

The  following  safe  values  are  recommended  for  cast 
iron  and  cast  steel. 


Linear  velocity 
ft.  per  min. 

100 

300 

300 

600 

900 

1200 

1800 

2400 

Cast 
Cast 

Iron  
Steel.  .  .  . 

8000 
24000 

6000 
15000 

4800 
12000 

4000 
10000 

3000 
7500 

2400 
6000 

2000 
5000 

1700 
4250 

For  gears  used  in  hoisting  machinery  where  there  is 
slow  speed  and  liability  of  shocks  a  writer  in  the 
American  Machinist  recommends  smaller  values  of  S 
than  those  given  above  *  and  proposes  the  following 
for  four  different  metals  : 

*  American  Machinist,  Feb.  16, 1905. 


EXPERIMENTAL  DATA.  153 


Linear  velocity 
ft.  per  min. 


Gray  Iron 


Gun  Metal 


Cast  Steel. 
Mild  Steel, 


100   200   300   600   900  1200  1800  2400 


4800  4200  3840  3200  2400  1920  1600  1360 

7200  6300  5760  4800  3600  2880  2400  2040 

9600  8400  7680  6400  4800  3840  3200  2720 

12000  10500  9600  8000  6000  4800  4000  3400 


The  experiments  described  in  the  next  article  show 
that  the  ultimate  values  of  S  are  much  less  than  the 
transverse  strength  of  the  material  and  point  to  the 
need  of  large  factors  of  safety. 

75.  Experimental  Data.  In  the  American  Machinist 
for  Jan.  14,  1897,  are  given  the  actual  breaking  loads 
of  'gear  teeth  which  failed  in  service.  The  teeth  had 
an  average  pitch  of  about  5  inches,  a  breadth  of  about 
18  inches  and  the  rather  unusual  velocity  of  over 
2000  ft.  per  minute.  The  average  breaking  load  was 
about  15000  Ib.  there  being  an  average  of  about  50 
teeth  on  the  pinions.  Substituting  these  values  in 
(88)  and  solving  we  get 

£=1575  Ib. 

This  very  low  value  is  to  be  attributed  to  the  con- 
dition of  pressure  on  one  corner  noted  in  Art.  73. 
Substituting  in  formula  for  such  a  case. 


This  all  goes  to  show  that  it  is  well  to  allow  large 
factors  of  safety  for  rough  gears,  especially  when  the 
speed  is  high. 

Experiments  have  been  made  on  the  static  strength 


154  MACHINE  DESIGN. 

of  rough  cast  iron  gear  teeth  at  the  Case  School  of 
Applied  Science  by  breaking  them  in  a  testing  machine. 
The  teeth  were  cast  singly  from  patterns,  were  two 
pitch  and  about  6  inches  broad.  The  patterns  were 
constructed  accurately  from  templates  representing 
15  deg.  involute  teeth  andcycloidal  teeth  drawn  with  a 
describing  circle  one-half  the  pitch  circle  of  15  teeth  ; 
the  proportions  used  were  those  given  for  standard  cut 
gears. 

There  were  in  all  41  cycloidal  teeth  of  shapes  cor- 
responding to  wheels  of  15-24-36-48-72-120  teeth  and 
a  rack.  There  were  28  involute  teeth  corresponding  to 
numbers  above  given  omitting  the  pinion  of  15  teeth. 

The  pressure  was  applied  by  a  steel  plunger  tangent 
to  the  surface  of  tooth  and  so  pivoted  as  to  bear  evenly 
across  the  whole  breadth.  The  teeth  were  inclined  at 
various  angles  so  as  to  vary  the  obliquity  from  0  to  25 
deg.  for  the  cycloidal  and  from  15  deg.  to  25  deg.  for 
the  involute.  The  point  of  application  changed  accord- 
ingly from  the  pitch  line  to  the  crest  of  the  tooth. 
From  these  experiments  the  following  conclusions  are 
drawn  : 

1.  The  plane  of  fracture  is  approximately  parallel  to 
line  of  pressure  and  not  necessarily  at  right  angles  to 
radial  line  through  center  of  tooth. 

2.  Corner  breaks  are  likely  to  occur  even  when  the 
pressure  is  apparently  uniform  along  the  tooth.    There 
were  fourteen  such  breaks  in  all. 

3.  With  teeth  of  dimensions  given,   the  breaking 
pressure  per  tooth  varies  from  25000  Ib.  to  50000  Ib. 
for  cycloids  as  the  number  of  teeth  increases  from  15 
to  infinity  ;  the  breaking  pressure  for  involutes  of  the 
same  pitch  varies  from  34000  Ib.  to  80000  Ib.  as  the 
number  increases  from  24  to  infinity. 


TEETH  OF  BEVEL  GEARS.  155 

4.  With  teeth  as  above  the  average  breaking  pres- 
sure varies  from  50000  Ib.  to  26000   Ib.  in  the  cycloids 
as  the  angle  changes  from  0  deg.  to  25  deg.  and  the 
tangent  point  moves  from  pitch  line  to  crest  ;  with 
involute  teeth  the  range  is  between  64000  and  39000  Ib. 

5.  Reasoning  from  the  figures  just  given,  rack  teeth 
are  about  twice  as  strong  as  pinion  teeth  and  involute 
teeth  have  an  advantage  in  strength  over  cycloidal  of 
from  forty  to  fifty  per  cent.     The  advantage  of  short 
teeth   in  point   of  strength   can   also  be  seen.      The 
modulus  of  rupture  of  the  material  used  was  about 
36000  Ib.     Values  of  S  calculated  from  Lewis'  formula 
for  the  various  tooth  numbers  are  quite  uniform  and 
average  about  40000  Ib.  for  cycloidal  teeth.     Involute 
teeth  are  to-day  generally  preferred  by  manufacturers. 
William  Sellers  &  Co.  use  an  obliquity  of  20  deg.  in- 
stead of  14 J  or  15  deg.  the  usual  angle. 

76.  Teeth  of  Bevel  Gears.  There  have  been  many 
formulas  and  diagrams  proposed  for  determining  the 
strength  of  bevel  gear  teeth,  some  of  them  being  very 
complicated  and  inconvenient.  It  will  usually  answer 
every  purpose  from  a  practical  standpoint,  if  we  treat 
the  section  at  the  middle  of  the  breadth  of  such  a  tooth 
as  a  spur  wheel  tooth  and  design  it  by  the  foregoing 
formulas.  The  breadth  of  the  teeth  of  a  bevel  gear 
should  be  about  one-third  of  the  distance  from  the  base 
of  the  cone  to  the  apex. 

One  point  needs  to  be  noted  ;  the  teeth  of  bevel 
gears  are  stronger  than  those  of  spur  gears  of  the 
same  pitch  and  number  of  teeth  since  they  are  developed 
from  a  pitch  circle  having  an  element  of  the  normal 
cone  as  a  radius.  To  illustrate,  we  will  suppose  that 
we  are  designing  the  teeth  of  a  miter  gear  and  that 


156  MACHINE  DESIGN. 

the  number  of  teeth  is  32.  In  such  a  gear  the  element 
of  normal  cone  is  j/  2  times  the  radius.  The  actual 
shape  of  the  teeth  will  then  correspond  to  those  of  a 
spur  gear  having  32  j/  2=45  teeth  nearly. 

NOTE. — In  designing  the  teeth  of  gears  where  the 
number  is  unknown,  the  approximate  dimensions  may 
first  be  obtained  by  formula  (84)  or  (85)  and  then  these 
values  corrected  by  using  Lewis'  formula. 

PROBLEMS. 

1.  The  drum  of  a  hoist  is  8  in.  in  diameter  and  makes  5 
rev.  per  minute.     The  diameter  of  gear  on  the  drum  is  36 
inches  and  of  its  pinion  6  in.     The  gear  on  the  counter  shaft 
is  24  in.  in  diameter  and  its  pinion  is  6  in.  in  diameter.    The 
gears  are  all  cut. 

Calculate  the  pitch  and  number  of  teeth  of  each  gear,  as- 
suming a  load  of  one  ton  on  drum  chain  and  $=6000.  Also 
determine  the  horse-power  of  the  machine. 

2.  Calculate   the  pitch   and  number  of  teeth   of  a  cut  cast 
steel  gear  10  in.  in  diameter,  running  at  250  rev.  per  min. 
and  transmitting  20  HP. 

3.  A  cast-iron  gear  wheel  is  30  ft.  6|  in.  in  pitch  diameter 
and  has  192  teeth,  which  are  machine-cut  and  30  in.  broad. 

Determine  the  circular  and  diameter  pitches  of  the  teeth  and 
the  horse-power  which  the  gear  will  transmit  safely  when 
making  12  rev.  per  min. 

4.  A  two  pitcli  cycloidal  tooth,  6  in.  broad,  72  teeth  to  the 
wheel,   failed  under  a  load  of  38000  Ib.     Find  value  of  S  by 
Lewis'  formula. 

5.  A  vertical  water-wheel  shaft  is  connected  to  horizontal 
head  shaft  by  cast  iron  gears  and  transmits  150  HP.      The 
water-wheel  makes  200  rev.  per  min.  and  the  head  shaft  100. 

Determine  the  dimensions  of  the  gears  and  teeth  if  the  latter 
are  approximately  two  pitch. 

6.  Work  Problem  1,  using  short  teeth  instead  of  standard. 


HIM  AND  ARMS.  157 

•77.  Rim  and  Arms.  The  rim  of  a  gear,  especially 
if  the  teeth  are  cast,  should  have  nearly  the  same 
thickness  as  the  base  of  tooth,  to  avoid  cooling  strains. 

It  is  difficult  to  calculate  exactly  the  stresses  on 
the  arms  of  the  gear,  since  we  know  so  little  of  the 
initial  stress  present,  due  to  cooling  and  contraction. 
A  hub  of  unusual  weight  is  liable  to  contract  in  cooling 
after  the  arms  have  become  rigid  and  cause  severe 
tension  or  even  fracture  at  the  junction  of  arm  and 
hub. 

A  heavy  rim  on  the  contrary  may  compress  the  arms 
so  as  actually  to  spring  them  out  of  shape.  Of  course 
both  of  these  errors  should  be  avoided,  and  the  pattern 
be  so  designed  that  cooling  shall  be  simultaneous  in 
all  parts  of  the  casting. 

The  arms  of  spur  gears  are  usually  made  straight 
without  curves  or  taper,  and  of  a  flat,  elliptical  cross- 
section,  which  offers  little  resistance  to  the  air.  To 
support  the  wide  rims  of  bevel  gears  and  to  facilitate 
drawing  the  pattern  from  the  sand,  the  arms  are  some- 
times of  a  rectangular  or  T  section,  having  the  greatest 
depth  in  the  direction  of  the  axis  of  the  gear.  For 
pulleys  which  are  to  run  at  a  high  speed  it  is  important 
that  there  should  be  no  ribs  or  projections  on  arms  or 
rim  which  will  offer  resistance  to  the  air.  Experiments 
by  the  writer  have  shown  this  resistance  to  be  serious 
at  speeds  frequently  used  in  practice. 

A  series  of  experiments  conducted  by  the  author  are 
reported  in  the  American  Machinist  for  Sept.  22, 1898, 
to  which  paper  reference  is  here  made. 

Twenty-four  pulleys  having  3^  inches  face  and 
diameters  of  16,  20  and  24  inches  were  broken  in  a 
testing  machine  by  the  pull  of  a  steel  belt,  the  ratio  of 
the  belt  tensions  being  adjusted  by  levers  so  as  to  be 


15$  MACHINE  DESIGN. 

two  to  one.  Twelve  of  the  pulleys  were  of  the  ordi- 
nary cast-iron  type  having  each  six  arms  tapering  and 
of  an  elliptic  section.  The  other  twelve  were  Medart 
pulleys  with  steel  rims  riveted  to  arms  and  having 
some  six  and  some  eight  arms.  Test  pieces  cast  from 
the  same  iron  as  the  pulleys  showed  an  average  modu- 
lus of  rupture  of  35800  for  the  cast-iron  and  50800  for 
the  Medart. 

In  every  case  the  arm  or  the  two  arms  nearest  the 
side  of  the  belt  having  the  greatest  tension,  broke  first, 
showing  that  the  torque  was  not  evenly  distributed  by 
the  rim.  Measurements  of  the  deflection  of  the  arms 
showed  it  to  be  from  two  to  six  times  as  great  on  this 
side  as  on  the  other.  The  buckling  and  springing  of 
the  rim  was  very  noticeable  especially  in  the  Medart 
pulleys. 

The  arms  of  all  the  pulleys  broke  at  the  hub  showing 
the  greatest  bending  moment  there,  as  the  strength  of 
the  arms  at  the  hub  was  about  double  that  at  the  rim. 
On  the  other  hand  some  of  the  cast  iron  arms  broke 
simultaneously  at  hub  and  rim,  showing  a  negative 
bending  moment  at  the  rim  about  one-half  that  at  the 
hub. 

The  following  general  conclusions  are  justified  by 
these  experiments  : 

(a)  The  bending  moments  on  pulley  arms  are  not 
evenly  distributed  by  the  rim,  but  are  greatest  next 
the  tight  side  of  belt. 

(b)  There  are  bending  moments   at  both  ends  of 
arm,  that  at  the  hub  being  much  the  greater,  the  ratio 
depending  on  the  relative  stiffness  of  rim  and  arms. 

The  following  rules  may  be  adopted  for  designing 
the  arms  of  cast  iron  pulleys  and  gears  : 

1.  Multiply  the  net  turning  pressure,  whether  caused 


SPROCKET  WHEELS  AND  CHAINS.  159 

by  belt  or  tooth,  by  a  suitable  factor  of  safety  and  by 
the  length  of  the  arm  in  inches.  Divide  this  product 
by  one-half  the  number  of  arms  and  use  the  quotient 
for  a  bending  moment.  Design  the  hub  end  of  arm  to 
resist  this  moment. 

2.  Make  the  rim  ends  of  arms  one-half  as  strong  as 
the  hub  ends. 

78.  Sprocket  Wheels  and  Chains.  Steel  chains  con- 
necting toothed  wheels  afford  a  convenient  means  of 
getting  a  positive  speed  ratio  when  the  axes  are  some 
distance  apart.  There  are  three  classes  in  common 
use,  the  block  chain,  the  roller  chain  and  the  so-called 
"  silent "  chain. 

Mr.  A.  Eugene  Michel  publishes  quite  a  complete 
discussion  of  the  design  of  the  first  two  classes  in 
Machinery  for  February,  1905,  and  reference  is  here 
made  to  that  journal. 

Block  chain  is  that  commonly  used  on  bicycles  and 
small  motor  cars,  so  named  from  the  blocks  with 
round  ends  which  are 
used  to  fill  in  between 
the  links.  The  sprocket 
teeth  are  spaced  to  a 
pitch  greater  than  that 
of  the  chain  links  and 
the  blocks  rest  on  flat 
beds  between  the  teeth, 
Fig.  73. 

Eoller  chains  have 
rollers  on  every  pin  and  have  inside  and  outside  links. 
The  sprocket  teeth  have  the  same  pitch  as  the  chain 
links,  the  rollers  fitting  circular  recesses  between  the 
sprockets,  Fig.  Y4. 


160 


MACHINE  DESIGN. 


The  most  serious  failing  of  the  chain  is  its  tendency 

to  stretch  with  use  so  that  the  pitch  becomes  greater 

than  that  of  the  sprocket  teeth. 

To  obviate  this  difficulty  in  a  measure  considerable 

clearance  should  be 
given  to  the  sprocket 
teeth  as  indicated  in 
Fig.  T4.  As  the  pitch 
of  the  chain  increases 
it  will  then  ride  higher 
upon  the  sprockets 
until  the  end  of  the 
tooth  is  reached.  The 
teeth  are  rounded  on 

their  side  faces,  that  they  may  easily  enter  the  gaps  in 

the  chain  and  have  side  clearance. 

Mr.  Michel  gives  the  following  values  for  the  tensile 

strength  of  chains  as  determined  by  actual  tests. 

EOLLER  CHAIN. 


Fig.  74. 


Pitch  inches 

i 

5 
"8 

i 

1 

H 

H 

If 

2 

Tensile 

Strength  Ib. 

1200 

1200 

4000 

6000 

9000 

12000 

19000 

25000 

BLOCK  CHAIN. 

1  inch  pitch  1200  to  2500  Ib. 
li    "     '    "  5000    " 

Mr.  Michel  further  recommends  a  factor  of  safety 
of  from  5  to  40  according  to  the  severity  of  the  condi- 
tions as  to  speed  and  shocks. 

The  tendency  is  to  use  short  links  and  double  or 
triple  width  chains  to  increase  the  rivet  bearing  sur- 


SILENT  CHAINS.  161 

face,  as  it  is  this  latter  factor  which  really  determines 
the  life  of  a  chain. 

Boiler  chains  may  be  used  up  to  speeds  of  1000  to 
1200  feet  per  minute. 

The  sprocket  should  be  so  designed  that  one  tooth 
will  carry  the  load  safely  with  the  pressure  near  the 
crest  since  these  conditions  obtain  as  the  chain  stretches. 
Use  values  of  S  as  in  Art.  74. 

79.  Silent  Chains,  The  weak  points  in  the  ordinary 
chain,  whether  it  be  made  with  blocks  or  rollers,  are 
the  rivet  bearings.  It  is  the  continual  wear  of  these, 
due  to  insufficient  area  and  lack  of  proper  lubrication, 
that  shortens  the  life  of  a  chain. 

The  so-called  "  silent- 
chain  "  with  rocker  bear- 
ings,- is  comparatively 
free  from  this  defect. 
Fig.  75  illustrates  the 
shapes  of  links,  rivets 
and  sprockets  for  this  >£. 

kind  of  chain  as  man- 
ufactured by  the  Morse  Chain  Company. 

The  chain  proper  is  entirely  outside  of  the  sprocket 
teeth  so  that  the  latter  may  be  continuous  across  the 
face  of  the  wheel,  save  for  a  single  guiding  groove  in 
the  center. 

Projections  on  the  under  side  of  the  links  engage 
with  the  teeth  of  the  sprocket,  E  being  the  point  of 
contact  for  the  driver  and  /  a  similar  point  for  the 
follower  when  the  rotation  is  as  indicated. 

Each  rivet  consists  practically  of  two  pins  called  by 
the  makers  the  rocker  pin  and  the  seat  pin.     Each  pin 
is  fastened  in  its  particular  gang  of  links  and  the 
n 


162 


MACHINE  DESIGN. 


relative  motion  is  merely  a  rocking  of  one  pin  on  the 
other  without  appreciable  friction. 

The  pins  are  of  hardened  tool  steel  with  softened 
ends.  The  combination  of  this  freedom  from  rubbing 
contact  with  the  adaptation  of  the  engaging  tooth 
profiles,  gives  a  chain  which  can  be  safely  run  at  high 
speeds  without  objectionable  vibration  or  appreciable 
wear. 

The  chains  can  be  made  of  almost  any  width  from 
one-half  inch  up  to  eighteen  inches,  the  width  de- 
pending upon  the  pitch  of  the  chain  and  the  power  to 
be  transmitted. 

The  following  are  the  working  loads  (and  limiting 
speeds)  of  chains  two  inches  in  width  and  of  different 
pitches,  taken  from  a  table  published  by  the  makers  : 


Pitch  in  inches 

1 

1 

3 

¥ 

.9 

1.2 

1.5 

Working  load 
in  pounds 

130 

190 

236 

380 

520 

760 

Limiting  Speed 

2000 

1600 

1200 

1100 

800 

600 

Eev.  per  min. 

The  number  of  teeth  in  the  small  sprocket  may  vary 
from  15  to  30  according  to  the  conditions. 

Assuming  17  teeth  and  the  number  of  revolutions 
given  in  the  above  table  the  speed  of  chain  would  be 
1420  feet  per  minute  for  the  |-  inch  pitch  and  1275  feet 
per  minute  for  the  1.5  inch. 

Chains  of  this  character  have  been  run  successfully 
at  2000  feet  per  minute. 


CRANKS  AND  LEVERS. 
PROBLEMS. 


163 


1.  Design  eight  arms  of  elliptic  section  for  a  gear  48  inches 
pitch  diameter,  to  transmit  a  pressure  on  tooth  of  800  pounds. 
Material,  cast  iron  having  a  working  transverse  strength  of 
6000  pounds  per  square  inch. 

2.  Two  sprocket  wheels  of  75  and  17  teeth  respectively  are 
to  transmit  twenty  horse-power  at  a  chain  speed  of  about  800 
feet  per  minute,  with  a  factor  of  safety  of  12 — 

Determine  the  proper  pitch  of  roller  chain,  the  pitch  diam- 
eters of  the  sprockets,  and  the  numbers  of  revolutions. 

3.  Suppose  that  in  Problem  2,  a  "silent"  chain  is  to  be 
used  and  the  chain  speed  increased  to  1200  feet  per  minute. 
Determine  the  proper  pitch  of  chain  to  be  used  if  the  width  of 
chain  is  3  inches.     Determine  diameters  and   revolutions  of 
sprockets  as  before. 

Cranks  and  Levers.  A  crank  or  rocker  arm  which 
is  used  to  transmit  a  continuous  or  reciprocating 
rotary  motion  is  in  the  condition  of  a  cantilever  or 
bracket  with  a  load  at  the  outer  end. 

If  the  web  of  the  crank  is  of  uniform  thickness 
theory  requires  that  its  profile  should  be  parabolic  for 
uniform  strength,  the  vertex  of  the  parabola  being  at 
the  load  point. 

A  convenient  approximation  to  this  shape  can  be 
attained  by  using  the  tangents  to  the  parabola  at 


Fig.  76.  ' 

points  midway  between  the  hub  and  the  load  point. 
See  Fig.  76,     The  crank  web  is  designed  of  the  right 


164:  MACHINE  DESIGN. 

thickness  and  breadth  to  resist  the  moment  at  AB, 
and  the  center  line  is  produced  to  Q,  making  PQ  =  \ 
PO. 

Straight  lines  drawn  from  Q  to  A  and  B  will  he 
tangent  to  the  parabola  at  the  latter  points  and  will 
serve  as  contour  lines  for  the  web. 

Assume  the  following  dimensions  in  inches  : 

Z  =  length  of  crank  ==  OP. 

t  —  thickness  of  web. 

h  =  breadth     "     "  =  AB. 

d  =  diameter  of  eye  =  cd. 

d1=         "         "   pin. 

b  =  breadth  of  eye. 

D  —  diameter  of  hub  =  CD. 

A=         "        "  shaft. 

B  =  breadth  of  hub. 

If  the  pressure  on  the  crank  pin  is  denoted  by  P 

PI 
then    will   the  moment  at   AB  be  —  and  the  equa- 

A 

tions  of  moments  for  the  cross-section  will  be  : 

Pl_Sth* 

2  "      6  [Bee  Formula  (3)] 

and  from  this  the  dimensions  at  AB  maybe  calculated. 
The  moment  at  the  hub  will  be  PI  and  will  tend  to 
break  the  iron  on  the  dotted  lines  CD.     The  equation 
of  moments  for  the  hub  is  therefore  : 

Pl=~-  (ZP-IV) 

From  this  equation  the  dimensions  of  the  hub  may 
be  calculated  when  Z)x  is  known.  The  eye  of  a  crank  is 
most  likely  to  break  when  the  pressure  on  the  pin  is 
along  the  line  OP,  and  the  fracture  will  be  along  the 
dotted  lines  cd.  The  bending  moment  will  be  P  mul- 


CRANKS  AND  LEVERS,  165 

tiplied  by  the  distance  from  center  of  pin  to  center  of 
eye  measured  along  axis  of  pin.  If  we  call  this  dis- 
tance Xj  then  will  the  equation  of  moments  be  : 


It  is  considered  good  practice  among  engine  builders 
to  make  the  values  of  x,  b  and  B  as  small  as  practicable, 
in  order  to  reduce  the  twisting  moment  on  the  web 
of  the  crank  and  the  bending  moment  on  the  shaft.  In 
designing  the  hub,  allowance  must  be  made  for  the 
metal  removed  at  the  key-  way. 

PROBLEM. 

Design  a  cast  steel  crank  for  a  steam  engine  having  a  cylin- 
der 12  by  30  inches  and  an  initial  steam  pressure  of  120  Ib. 
per  sq.  in.  of  piston.  The  shaft  is  6  inches  and  the  crank  pin 
3  inches  in  diameter.  The  distance  x  may  be  assumed  as  4 
inches.  Calculate, 

1.  Dimensions  of  web  at  AB. 

2.  Dimensions  of  hub  allowing  for  a  key  1  xf  inches. 

3.  Dimensions  of  eye  for  pin,  make  a  scale  drawing  in  ink 
showing  profile  of  crank  complete,  S  may  be  assumed  as  6,000 
Ib.  per  sq.  in. 


CHAPTEK  XI. 

FLY-WHEELS. 

81.  In  General.  The  hub  and  arms  of  a  fly-wheel 
are  designed  in  much  the  same  way  as  those  of  pulleys 
and  gears,  the  straight  arm  with  elliptic  section  being 
the  favorite.  The  rims  of  such  wheels  are  of  two 
classes,  the  wide,  thin  rim  used  for  belt  transmission 
and  the  narrow  solid  rim  of  the  generator  or  blowing 
engine  wheel.  Fly-wheels  up  to  eight  or  ten  feet  in 
diameter  are  usually  cast  in  one  piece  ;  those  from  ten 
to  sixteen  feet  in  diameter  may  be  cast  in  halves,  while 
wheels  larger  than  the  last  mentioned  should  be  cast 
in  sections,  one  arm  to  each  section. 

This  is  a  matter,  not  of  use,  but  of  convenience  in 
transportation. 

The  joints  between  hub  and  arms  and  between  arms 
and  rim  need  not  be  specially  considered  here,  since 
wheels  rarely  fail  at  these  points. 

The  rim  and  the  joints 
in  the  rim  cannot  be  too 
carefully  designed.  The 
smaller  wheel  cast  in  one 
piece  is  more  or  less  sub- 
ject to  stresses  caused  by 
shrinkage.  The  sectional 
Fig.  77.  '  wheel  is  generally  free 

from  such  stresses  but  is 
weakened  by  the  numerous  joints. 

Kim  joints  are  of  two  general  classes  according  as 
bolts  or  links  are  used  for  fastenings. 

166 


SAFE  SPEED  FOR  WHEELS. 


167 


"Wide,  thin  rims  are  usually  fastened  together  by 
internal  flanges  and  bolts  as  shown  in  Fig.  77,  while 
the  stocky  rims  of  the  fly-wheels  proper  are  joined 
directly  by  links  or  T—  head  "  prisoners  "  as  in  Fig.  78. 

As  will  be  shown 
later,  the  former  is  a 
weak  and  unreliable 
joint,  especially 
when  located  mid- 
way between  the 
arms. 

The  principal 
stresses  in  fly-wheel 
rims  are  caused  by  centrifugal  force. 

82.  Safe  Speed  for  Wheels.  The  centrifugal  force 
developed  in  a  rapidly  revolving  pulley  or  gear  pro- 
duces a  certain  tension  on  the  rim,  and  also  a  bending 
of  the  rim  between  the  arms.  We  will  first  investigate 
the  case  of  a  pulley  having  a  rim  of  uniform  cross 
section. 

It  is  safe  to  assume  that  the  rim  should  be  capable 
of  bearing  its  own  centrifugal  tension  without  assist- 
ance from  the  arms. 

Let      _D=mean  diameter  of  pulley  rim. 
t= thickness  of  rim. 
b= breadth  of  rim. 
w= weight  of  material  per  cu.  in. 
=  .26  Ib.  for  cast-iron. 
=  .28  Ib.  for  wrought  iron  or  steel. 
n= number  of  arms. 
N=  number  rev.  per  min. 
v= velocity  of  rim  in  ft.  per  sec. 


108  MACHINE  DESIGN. 

First  let  us  consider  the  centrifugal  tension  alone. 
The  centrifugal  pressure  per  square  inch  of  concave 
surface  is 


„  (  \ 

p=  -  -        ....         (a) 

gr 
where  W  is  the  weight  of  rim  per  square  inch  of  con- 

cave surface  =wt,  and  r=  radius  in  feet  =  — 

OTC 

The  centrifugal  tension  produced  in  the  rim  by  this 
force  is  by  formula  (13) 


Substituting  the  values  of  p,  W  and  r  and  reducing  : 

8=™a>*    ....    (89) 
and  v=>fef  •     •     •     •    (90) 

For  an  average  value  of  w=.27,  (89)  reduces  to 


a  convenient  form  to  remember. 

The  corresponding  values  of  S  for  dry  wood  and  for 
leather  would  be  nearly  : 

Wood  S 


Leather  S=~ 

oO 

If  we  assume  S  as  the  ultimate   tensile  strength, 
16500  Ibs.  for  cast-iron  in  large  castings  and  60000  Ibs. 
for  soft  steel,  then  the  bursting  speed  of  rim  is  : 
for  a  cast-iron  wheel        ^=406  ft.  per  sec.        .        (91) 
and  for  steel  rim  v=7T5  ft.  per  sec.        .        (92) 

and  these  values  may  be  used  in  roughly  calculating 
the  safe  speed  of  pulleys. 


SAFE  SPEED  FOR  WHEELS.  169 

It  has  been  shown  by  Mr.  James  B.  Stanwood,  in  a 
paper  read  before  the  American  Society  of  Mechanical 
Engineers,*  that  each  section  of  the  rim  between  the 
arms  is  moreover  in  the  condition  of  a  beam  fixed  at 
the  ends  and  uniformly  loaded. 

This  condition  will  produce  an  additional  tension 
on  the  outside  of  rim.  The  formula  for  such  a  beam 
when  of  rectangular  cross-section  is 

Wl_  Sbd*  ,,. 

"12""     6 

W  in  this  case  is  the  centrifugal  force  of  the  fraction 
of  rim  included  between  two  arms. 

The  weight  of  this  fraction  is  *Dbtw  and  its  cen- 


trifugal  force  W=X         or 


n         g  gn 

Also  i= 


n 

Substituting  these  values  in  (b)  and  solving  for  S  : 
o     o  r^oDwv2  /  x 

~w  .......  '  ' 

If  w  is  given  an  average  value  of  .27  then 

8—jg-  nearly  ......  (d) 

and  the  total  value  of  the  tensile  stress  on  outer  sur- 
face of  rim  is 


Solving  for  v : 

T~jT~ 
v=^l~p     T  . (94) 

In  a  pulley  with  a  thin  rim  and  small  number  of 


*  See  Trans.  A.  S.  M.  E.  Vol.  XIV. 


170  MACHINE  DESIGN, 

arms,  the  stress  due  to  this  bending  is  seen  to  be  con- 
siderable. 

It  must,  however,  be  remembered  that  the  stretching 
of  the  arms  due  to  their  own  centrifugal  force  and  that 
of  the  rim  will  diminish  this  bending.  Mr.  Stan  wood 
recommends  a  deduction  of  one-half  from  the  value  of 
S  in  (d)  on  this  account. 

Prof.  Gaetano  Lanza  has  published  quite  an  elab- 
orate mathematical  discussion  of  this  subject.  (See 
Vol.  XVI.  Trans.  Am.  Soc.  Mech.  Engineers.)  He 
shows  that  in  ordinary  cases  the  stretch  of  the  arms 
will  relieve  more  than  one-half  of  the  stress  due  to 
bending,  perhaps  three-quarters. 

83.  Experiments  on  Fly- Wheels.  In  order  to  de- 
termine experimentally  the  centrifugal  tension  and 
bending  in  rapidly  revolving  rims,  a  large  number  of 
small  fly-wheels  have  been  tested  to  destruction  at  the 
Case  School  laboratories.  In  all  ten  wheels,  fifteen 
inches  in  diameter  and  twenty-three  wheels  two  feet  in 
diameter  have  been  so  tested.  An  account  of  some  of 
these  experiments  may  be  found  in  Trans.  Am.  Soc. 
Mech.  Eng.  Vol.  XX.  The  wheels  were  all  of  cast- 
iron  and  modeled  after  actual  fly-wheels.  Some  had 
solid  rims,  some  jointed  rims  and  some  steel  spokes. 

To  give  to  the  wheels  the  speed  necessary  for  de- 
struction, use  was  made  of  a  Dow  steam  turbine  capa- 
ble, of  being  run  at  any  speed  up  to  10000  revolutions 
per  minute.  The  turbine  shaft  was  connected  to  the 
shaft  carrying  the  fly-wheels  by  a  brass  sleeve  coup- 
ling loosely  pinned  to  the  shafts  at  each  end  in  such  a 
way  as  to  form  a  universal  joint,  and  so  proportioned 
as  to  break  or  slip  without  injuring  the  turbine  incase 
of  sudden  stoppage  of  the  fly-wheel  shaft. 


EXPERIMENTS  ON  FLY-WHEELS. 


171 


One  experiment  with  a  shield  made  of  two-inch 
plank  proved  that  safety  did  not  lie  in  that  direction, 
and  in  succeeding  experiments  with  the  fifteen  inch 
wheels  a  "bomb-proof  constructed  of  6X12  inch  white 
oak  was  used.  The  first  experiment  with  a  twenty- 
four  inch  wheel  showed  even  this  to  be  a  flimsy  contri- 
vance. In  subsequent  experiments  a  shield  made  of 
12x12  inch  oak  was  used.  This  shield  was  split  re- 
peatedly and  had  to  be  re-enforced  by  bolts. 

A  cast  steel  ring  about  four  inches  thick  lined,  with 
wooden  blocks  and  covered  with  three  inch  oak  plank- 
ing, was  finally  adopted. 

The  wheels  were  usually  demolished  by  the  ex- 
plosion. No  crashing  or  rending  noise  was  heard, 
only  one  quick,  sharp  report,  like  a  musket  shot. 

The  following  tables  give  a  summary  of  a  number 
of  the  experiments. 

TABLE  XXVI. 

FIFTEEN  INCH  WHEELS. 


Bursting  Speed. 

Centrifugal 

No. 

Tension 

v* 

Remarks. 

Rev. 

Feet  per 

i~n 

per  Minute. 

Second  =v. 

1U 

1 

6,525 

430 

18,500 

Six  arms. 

-...-2 

6,525 

430 

18,500 

Six  arms. 

3 

6,035 

395 

15,600 

Thin  rim. 

4 

5,872 

380 

14,400 

Thin  rim. 

5 

2,925 

192 

3,700 

Joint  in  rim. 

6 

5,600* 

368 

13,600 

Three  arms. 

7 

6,198 

406 

16,500 

Three  arms. 

8 

5,709 

368 

13,600 

Three  arms. 

9 

5,709 

365 

13,300 

Thin  rim. 

10 

5,709 

361 

13,000 

Thin  rim. 

Doubtful. 


MACHINE  DESIGN. 


TABLE  XXVII. 

TWENTY-FOUR  INCH  WHEELS. 


Shape  and  Size  of  Rim. 

Weight 
f\f 

d 
fc 

Diam- 

Breadth 

Depth 

Area 

Wheel, 

eter 
Inches. 

Inches. 

Inches. 

Sq. 
Inches. 

Style  of  Joint. 

Pounds 

11 

24 

3i 

1.5 

3.18 

Solid  rim.              • 

75.25 

12 

24 

*rV 

.75 

3.85 

Internal  flanges,bolted 

93. 

13 

24 

4 

.75 

3.85 

<(                  u                  u 

91.75 

14 

24 

4 

.75 

3.85 

«               ««               « 

95. 

15 
16 

24 
24 

a 

.75 

2.1 

3.85 
2.45 

«                  «                  « 

Three  lugs  and  links. 

94.75 
65.1 

17 

24 

1.2 

2.1 

2.45 

Two  lugs  and  links. 

65. 

TABLE  XXVIII. 

FLANGES  AND  BOLTS. 


FLANGES. 

BOLTS. 

No. 

Thickness. 

Effective 
Breadth. 

Effective 
Area. 

No.  to  each 

Diameter. 

Total 
Tensile 

Inches. 

Inches. 

Inches. 

Joint. 

Inches. 

Strength. 
Pounds. 

12 

f 

2.8 

1.92 

4 

T'V 

16,000 

13 

2.75 

1.89 

4 

A 

16,000 

14 
15 

ij 

2.75 
2.5 

2.58 
2.34 

4 

4 

t 

16,000 
20,000 

BY  TESTING  MACHINE. 

Tensile  strength  of  cast-iron  =19,600  pounds  per  square  in. 
Trans  verse  strength  of  cast-iron =46, 600  pounds  per  square  in. 
Tensile  strength  of  T5^  bolts        =4,000  pounds. 
Tensile  strength  of  f  bolts         =5,000  pounds. 


EXPERIMENTS  ON  FLY-WHEELS. 

TABLE  XXVIX. 

FAILURE  OP  FLANGED  JOINTS. 


173 


• 

Bursting 

Cent. 

rfi 

5   w 

Speed. 

Tension. 

s  s 

j-j  02 

*     o 

No. 

"^ 

i     PH 

Rev. 

Ft.  per 

Per 

REMARKS. 

2   1 

i  i 

3   .2 

per 

Sec. 

Sq.  In. 

Total 

<ri     o* 

w   g 

5  "o 

V7 

Lbs. 

CC 

1 

H    pq 

Min. 

=« 

10 

11 

3.18 

3,672 

385 

14,800 

47,000 

Solid  rim. 

13 
14 

3.85 

3.85 
3.85 

1.92 
1.89 

2.58 

16,000 
16,000 
16,000 

Flange  broke. 
Flange  broke. 
Bolts  broke. 

1,760 
1,875 

184 
196 

3,400 
3,850 

13,100 
14,800 

15 

3.85 

2.34 

20,000 

1,810 

190 

3,610 

13,900 

Flange  broke. 

TABLE  XXX. 

LINKED  JOINTS. 

LUGS. 

LINKS. 

RIM. 

No. 

.< 

t 

a* 
i—  i 

« 

^ 

rf 

of 

Max. 

Net 

,c  •& 
•a  c 

A  "o 

|i 

TJ  05 

0    oi 
M    <u 

Area, 

Area, 

2 

a 

3 

1 

3 

Q)   S-    O 

0   43 

gi 

§   51 

Sq.   Ins. 

Sq.  Ins. 

16 

.45 

1.0 

.45 

3 

.57 

.327 

.186 

2.45 

1.98 

17 

.44 

.98 

.43 

2 

.54 

.380 

.205 

2.45 

1.98 

BY  TESTING  MACHINE. 

Tensile  strength  of  cast-iron  =19,600. 
Transverse  strength  of  cast-iron =40, 400. 
Av.  tensile  strength  of  each  link=:  10,180. 


174 


MACHINE  DESIGN. 


TABLE  XXXI. 

FAILURE  OF  LINKED  JOINTS. 


No. 

ngth  of  Links, 
Pounds. 

ngth  of  Rirn, 
Pounds. 

BURSTING 
SPEED. 

CENT.  TENSION. 

REMARKS. 

Rev. 
per 

Ft.  per 
See. 

Per 
Sq.  In. 

Total. 

g 

1 

Min. 

=v 

10 

16 
17 

30,540 
20,360 

38,800 
38,800 

3,060 
2,750 

320 

290 

10,240 

8,410 

25,100 
20,600 

Rim  broke. 
Lugs  and  Rim 
broke. 

The  flanged  joints  mentioned  had  the  internal  flanges 
and  bolts  common  in  large  belt  wheel  rims  while  the 
linked  joints  were  such  as  are  common  in  fly-wheels 
not  used  for  belts. 

*  Subsequent  experiments  have  given  approximately 
the  same  results  as  those  just  detailed.  The  highest 
velocity  yet  attained  has  been  424  feet  per  second  ;  this 
is  in  a  solid  cast-iron  rim  with  numerous  steel  spokes. 
The  average  bursting  velocity  for  solid  cast  rims  with 
cast  spokes  is  400  feet  per  second. 

Wheels  with  jointed  rims  burst  at  speeds  varying 
from  190  to  250  feet  per  second,  according  to  the  style 
of  joint  and  its  location.  The  following  general  con- 
clusions seem  justified  by  these  tests. 

1.  Fly-wheels  with  solid  rims,  of  the  proportions 
usual  among  engine  builders  and  having  the  usual 
number  of  arms,  have  a  sufficient  factor  of  safety  at  a 
rim  speed  of  100  feet  per  second  if  the  iron  is  of  good 
quality  and  there  are  no  serious  cooling  strains. 

In  such  wheels  the  bending  due  to  centrifugal  force 
is  slight,  and  may  safely  be  disregarded. 

*  See  Trans.  Am.  Soc.  Mech.  Eng.,  Vol.  XXIII. 


EXPERIMENTS  ON  FLY-WHEELS. 

2.  Rim  joints  midway  between  the  arms  are  a  serious 
defect  and  reduce  the  factor  of  safety  very  materially. 
Such  joints  are  as  serious  mistakes  in  design  as  would 
be  a  joint  in  the  middle  of  a  girder  under  a  heavy  load. 

3.  Joints  made  in  the  ordinary  manner,  with  internal 
flanges  and  bolts,  are  probably  the  worst  that  could  be 
devised  for  this  purpose.     Under  the  most  favorable 
circumstances   they  have  only  about  one-fourth  the 
strength  of  the  solid  rim  and  are  particularly  weak 
against  bending. 

See  Fig.  79,  which  shows  the  opening  of  such  a  joint 
and  the  bending  of  the  bolts* 

In  several  joints  of  this  character,  on  large  fly- 
wheels, calculation  has  shown  a  strength  less  than  one- 
fifth  that  of  the  rim. 

4.  The  type  of  joint  known  as  the  link  or  prisoner 
joint  is  probably  the  best  that  could  be  devised  for 
narrow  rimmed  wheels  not  intended  to  carry  belts,  and 
possesses,  when  properly  designed,   a  strength  about 
two-thirds  that  of  the  solid  rim. 

In  1902-04  experiments  on  four-foot  pulleys  were 
conducted  by  the  writer,  and  the  results  published.* 

A  cast-iron,  whole  rim  pulley  48  inches  in  diameter, 
burst  at  1100  rev.  per  min.  or  a  linear  speed  of  230  ft. 
per  sec.,  the  rupture  being  caused  by  a  balance  weight 
of  3£  pounds  which  had  been  riveted  inside  the  rim  by 
the  makers.  The  centrifugal  force  of  this  weight  at 
1100  rev.  per  min.  was  2760  Ib. 

A  cast  iron  split  pulley  of  the  same  dimensions  burst 
at  a  speed  of  about  600  rev.  per  min.,  or  a  linear  speed 
of  only  125  ft.  per  sec. 

The  failure  was  due  to  the  unbalanced  weight  of  the 


*  Trans,  Am,  SOQ.  Mech.,  Eng.,  Vol.  XXVI, 


176  MACHESE  DESIGN. 

joint  flanges  and  bolts  which  were  located  midway  be- 
tween the  arms.  Such  a  pulley  is  not  safe  at  high  belt 
speeds. 

84.  Wooden  Pulleys.  Experiments  on  the  bursting 
strength  of  wooden  pulleys  were  conducted  at  the  Case 
School  laboratories  in  1902-3  under  the  writer's  direc- 
tion.* 

These  are  of.  some  interest  in  view  of  the  use  of  this 
material  for  fly-wheel  rims.  As  rioted  in  Art.  82,  the 
tensile  stress  in  wood  due  to  the  centrifugal  force  is 
only  y1^  that  of  cast-iron  under  similar  circumstances. 
Assuming  the  tensile  strength  of  the  wood  to  be  10000 
Ibs.  per  sq.  in.,  and  substituting  this  value  in  theequa- 

v2 
tion  S  =          we  have  the  bursting  speed  of  a  wooden 


pulley  v=WOO  ft.  per  sec.  nearly. 

This  for  wood  without  joints. 

The  24:  inch  pulleys  tested  had  wood  rims  glued  up  in 
the  usual  manner  and  jointed  at  two  opposite  points. 
The  wheels  burst  at  speeds  varying  from  1700  to  2450 
rev.  per  min.,  or  linear  rim  speeds  varying  from  178  to 
257  ft.  per  sec.,  thus  comparing  favorably  with  cast 
iron  split  pulleys.  The  rims  usually  failed  at  the 
points  where  the  arms  were  mortised  in,  and  the  stif- 
fening braces  at  these  points  did  more  harm  than  good. 
A  wooden  pulley  with  solid  rim  and  web  remained  in- 
tact at  4450  rev.  per  min.,  or  467  ft.  per  sec.,  a  higher 
speed  than  that  of  any  cast-iron  pulley  tried. 

85.  Rims  of  Cast-iron  Gears.  A  toothed  wheel  will 
burst  at  a  less  speed  than  a  pulley  because  the  •  teeth 

*  Machinery,  N.  Y.,  Aug.,  1905, 


ROTATING  DISCS.  177 

increase  the  weight  and  therefore  the  centrifugal  force 
without  adding  to  the  strength. 

The  centrifugal  force  and  therefore  the  stresses  due 
to  the  force  will  be  increased  nearly  in  the  ratio  that 
the  weight  of  rim  and  teeth  is  greater  than  the  weight 
of  rim  alone. 

This  ratio  in  ordinary  gearing  varies  from  1.5  to 
1.7.  We  will  assume  1.6  as  an  average  value.  Neg- 
lecting bending  we  now  have  from  equation  (89) 

o    16x12w?;2  ^_l$.2wv2  .......  (95) 

g  g 

and  t;=\l9^7 

=  326.2  ft.  per  second  .     .     .   (96) 
Including  bending 

' 


As  the  transverse  strength  of  cast  iron  by  experi- 
ment is  about  double  the  tensile  strength,  a  larger 
value  of  S  may  be  allowed  in  formulas  (93)  (94)  and 
(97.) 

In  built  up  wheels  it  is  better  to  have  the  joints  come 
near  the  arms  to  prevent  the  tendency  of  the  bending 
to  open  the  joints,  and  the  fastenings  should  have  the 
same  tensile  strength  as  the  rim  of  the  wheel. 

86.  Rotating  Discs.  The  formulas  derived  in  Art. 
82  will  only  apply  in  the  case  of  thin  rims  and  cannot 
be  used  for  discs  or  for  rims  having  any  considerable 
depth.  The  determination  of  the  stresses  in  a  rotating 
disc  is  a  complicated  and  difficult  problem,  if  the  ma- 
terial is  regarded  as  perfectly  elastic. 

A  rational  solution  of  this  problem  may  be  found  in 
Storlola's  Steam  Turbines,  pp.  157-69,  For  the  pur- 


178 


MACHINE  DESIGN. 


poses  of  this  treatise  an  approximate  solution  is  pre- 
ferred, the  elasticity  of  the  metal  being  neglected. 
This  method  of  treatment  is  much  simpler,  and  as  the 
metals  used  are  imperfectly  elastic  (especially  the  cast 
metals)  the  results  obtained  will  probably  be  as  reliable 
as  any — for  practical  use. 

The  following  discussion  is  an  abstract  of  one  given 
by  Mr.  A.  M.  Levin  in  the  American  Machinist  *  the 
notation  being  changed  somewhat. 


87.  Plain  Discs. 
Let  Fig.  80  represent 
a  ring  of  uniform 
thickness  t,  having 
an  external  diameter 
D  and  an  internal 
diameter  d,  all  in 
inches. 

Let  v— external  ve- 
locity in  feet  per  sec- 
ond. 


Fig.  80. 


Let    a = angular  velocity  = 

JLS 

r= radius  to  center  of  gravity  of  half  ring 

in  feet. 

w= weight  of  metal  per  cubic  inch. 
The  value  of  r  for  a  half-ring  is  easily  proved  to  be  : 

c\  r\3  -73 

in  inches 


or 


American  Machinist,  Oct.  20, 1904, 


ROTATING  DISCS.  179 

The  weight  of  the  half-ring  is  : 


o 

and  its  centrifugal  force  : 


Substituting  for  a  its  value  in  terms  of  v  : 

c^JwvW^F)     ........     (99) 

Now  if  we  assume  the  stress  on  the  area  at  AS  due 
to  the  centrifugal  force  to  be  uniformly  distributed  : 
(and  here  lies  the  approximation)  then  will  the  tensile 
stress  on  the  section  be 


o         C  ,1Anv 

=(D=d)t==          ~z- 


For  a  solid  disc  : 

or        4:WV'  fiM\ 

&d=0=-—     .......     .     .     (101) 

i/ 

For  a  thin  ring  : 

S~-^f      ........      (102) 

on  the  same  as  in  equation  (89). 

If  the  metal  be  perfectly  elastic,  Stodola's  formulas 

O  7/77  9^ 

give  $=  -  as  the  stress  near  the  center  when  d  ap- 

y 

proaches  0—  or  more  than  twice  the  value  given  in 
(101).  In  view  of  the  imperfect  elasticity  of  the  metals 
used  the  true  value  will  probably  be  between  these  two, 
This  value  should  be  determined  by  experiment, 


180 


MACHINE  DESIGN. 


88.  Conical  Discs.  Let  Fig.  81  represent  a  ring 
whose  thickness  varies  uniformly  from  the  inner  to  the 
outer  circumference  and  whose  dimensions  are  as  fol- 
lows : 

D=  outer  diameter  in  inches. 

d=  inner  diameter 

6=  breadth  of  ring  at  inner  circumference. 

m=  tan  gent  of  angle  of  slant  CAD. 

Then  m= 


m 


By  cutting  the  ring  into  slices  perpendicular  to  the 


/ 

A( 

ll 

h 

D 

V 

\ 

\ 

\ 

|l 
|l 

/; 
/ 

Fig.  81. 

axis,  finding  the  centrifugal  force  for  each  slice  and 
then  integrating  between  D  and  <i,  the  centrifugal 
force  of  the  half-ring  is  found  to  be  : 


mgD2 
The  area  on  the  line  A  B  to  resist  the  centrifugal 

force  is  :  (  and  g- 


BURSTING  SPEEDS.  181 

When  d=0  : 

SJ*^.     . (105) 

y 

or  a  stress  one-half  that  of  a  plain  flat  disc. 

89.  Discs  with  Logarithmic  Profile.  A  form  of 
disc  sometimes  used  for  steam  turbines  consists  of  a 
solid  of  revolution  generated  by  a  curve  of  the 
equation 

y=a  log  | 

revolving  around  the  x— axis. 

Mr.  Levin  investigates  two  curves  of  this  character  : 

x 
y=log  x  and  y=2  log  ^ 

and  finds  the  stresses  to  be  respectively  : 


When  a=b  #=1.5^ .(106) 

»/ 


When  a=\b  #  =  1.2— (107) 

The  general  equation  for  8  in  this  case  is  : 

#=96^.-^.      ....     .(108) 

and  in  deriving  the  formulas  (106)  and  (107)  D  is  as- 
sumed as  8a  and  as  9a  respectively. 

90.  Bursting  Speeds.  It  will  be  seen  that  all  the 
formulas  for  centrifugal  stress  may  be  reduced  to  the 
general  form  : 

S  =  k—.  .    .          .          .(109) 

9 

where  k  is  a  constant  depending  upon  the  shape  of  the 
rotating  body. 


182 


MACHINE  DESIGN. 


The  following  table  gives  the  values  of  v=  A(]T— > 

bursting  speed  of  iron  in  feet  per  second,  for  different 
materials  and  different  shapes. 

TABLE  XXXII. 
BURSTING  SPEEDS  IN  FEET  PER  SECOND. 


Metal. 

S-g 

*& 

§.2 

Tensile 
Strength. 

Values  of  v. 

Thin 
Ring. 

Perforated 
Disc 
(Stodola). 

Flat 
Disc. 

Taper 
Disc. 

Logar- 
ithmic 
Disc. 

w 

*=!» 

,=9 

fc=4 

fc=a 

i=w 

Cast  Iron. 

.026 
.0315 
.028 

18000 
60000 
60000 

430 
715 
760 

500 

825 
880 

745 
1240 
1315 

1050 
1750 
1860 

1215 
2050 
2140 

Manganese  Bronze 
Soft  Steel  

PROBLEMS. 

1.  Determine  bursting  speed  in  revolutions  per  minute,  of  a 
gear  48  inches  in  diameter  with  six  arms,  if  the  thickness  of 
rim  is  .75  inch. 

(1)  Considering  centrifugal  tension  alone. 

(2)  Including  bending  of  rim  due  to  centrifugal  force  as- 
suming that  f  the  stress  due  to  bending  is  relieved  by  the 
stretching  of  the  arms. 

2.  Design  a  link  joint  for  the  rim  of  a  fly-wheel,  the  rim 
being  8  in.  wide,  12  in.  deep  and  18  ft.  mean  diameter,  the 
links  to   have  a  tensile    strength   of   65000   Ib.    per  sq.    in. 
Determine  the  relative  strength  of  joint  and  the  probable 
bursting  speed. 


BURSTING  SPEEDS.  183 

3.  Discuss  the  proportions  of  one  of  the  following  wheels  in 
the  laboratory  and  criticise  dimensions. 

(a)  Fly-wheel.  Allis  engine. 

(b)  Fly-wheel,  Fairbanks  gas  engine. 

(c)  Fly-wheel,  air  compressor. 
(a)  Fly-wheel,  Ball  engine. 

(e)   Fly-wheel,  ammonia  compressor. 

4.  Determine  the  value  of  C  in  formula  (103)  by  calculation. 

5.  A  Delaval  turbine  disc  is  made  of  soft  steel  in  the  shape 
of  the  logarithmic  curve  without  any  hole  at  the  center. 
Determine  the  probable  bursting  speed  if  the  disc  is  8  inches  in 
diameter. 

6.  A  wheel  rim  is  made  of  cast  iron  in  the  shape  of  a  ring 
having  diameters  of  4£  feet  and  6  feet,  inside  and  outside. 
Determine  probable  bursting  speed. 

7.  Substitute  the  value  for  centrifugal  force  in  place  of 
internal  pressure  in  Barlow's  formula  (b)  Art.  12,  and  derive  a 

value  for  Sin  a  rotating  ring.     Test  this  for  d=~  and  compare 

A 

with  formulas  in  preceding  article. 


CHAFTER  XII. 

TRANSMISSION  BY  BELTS  AND  ROPES. 

91.  Friction  of  Belting.  The  transmitting  power  of 
a  belt  is  due  to  its  friction  on  the  pulley,  and  this  friction 
is  equal  to  the  difference  between  the  tensions  of  the 
driving  and  slack  sides  of  the  belt. 

Let  w  =  width  of  belt. 

Ji=  tension  of  driving  side. 
T2=  tension  of  slack  side. 
R  =  friction  of  belt. 


/= coefficient  of  friction  be- 
tween belt  and  pulley. 
0  =  arc  of  contact  in  circu- 
r.  82.  lar  measure. 

The  tension  T  at  any  part  of  the  arc  of  contact  is  in- 
termediate between  Ti  and  T2 . 

Let  AB  Fig.  82  be  an  indefinitely  short  element  of 
the  arc  of  contact,  so  that  the  tensions  at  A  and  B 
differ  only  by  the  amount  d T. 

dT  will  then  equal  the  friction  on  AB  which  we  may 
call  dR. 

Draw  the  intersecting  tangents  OTand  OT  to  rep- 
resent the  tensions  and  find  their  radial  resultant 
OP.  Then  will  OP  represent  the  normal  pressure  on 
the  arc  AB  which  we  will  call  P. 

<OTP=<ACB  =  dO 
.'.  P=TdO 

184: 


FRICTION  OF  BELTING. 


185 


The  friction  on  AB  is 

fP=fTd6 
or  dT=dR=fTd& 

fd0=dT 

T 

Integrating  for  the  whole  arc  0  : 


and 


The  average  value  of/  for  leather  belts  on  iron  pul- 
leys as  determined  by  experiment  is  /=0.2Y. 

If  we  denote  expression  (1—e—fO)  by  (7,  then  for  dif- 
ferent arcs  of  contact  C  has  the  following  values  : 


Arc  of 
Contact. 

90° 

110° 

130° 

150° 

180° 

210° 

240° 

C 

.345 

.404 

.458 

.506 

.571 

.627 

.676 

The  friction  or  force  transmitted  by  a  belt  per  inch 
of  width  is  then 

R=CTi (Ill) 

and  Ji   must  not   exceed   the    safe  working    tensile 
strength  of  the  material. 

A  handy  rule  for  calculating  belts  assumes  C=.5 
which  means  that  the  force  which  a  belt  will  transmit 


186  MACHINE  DESIGN. 

under    ordinary    conditions    is    one-half    its    tensile 
strength. 

92.  Strength  of  Belting.     The  strength  of  belting 
varies  widely  and  only  average  values  can  be  given. 
According  to  experiments  made  by  the  author  good  oak 
tanned  belting  has  a  breaking  strength  per  inch  of 
width  as  follows  : 

Single.  Double. 

Solid  leather 900  Ib.  1400  Ib. 

Where  riveted 600  Ib.  1200  Ib. 

Where  laced 350  Ib.  

Canvas  belting  has  approximately  the  same  strength 
as  leather.  Tests  of  rubber  coated  canvas  belts  4-ply, 
8  inches  wide,  show  a  tensile  strength  of  from  840  Ib. 
to  930  Ib.  per  inch  of  width. 

93.  Taylor's  Experiments.     The  experiments  of  Mr. 
F.  W.  Taylor,  as  reported  by  him  in  Trans.  Am.  Soc. 
Mech.  Eng.  Yol.  XV.  afford  the  most  valuable  data 
now  available  on  the  performance  of  belts  in  actual 
service. 

These  experiments  were  carried  on  during  a  period 
of  nine  years  at  the  Midvale  Steel  Works.  Mr.  Taylor's 
conclusions  may  be  epitomized  as  follows  : 

1.  Narrow  double  belts  are  more  economical  than 
single  ones  of  a  greater  width. 

2.  All  joints  should  be  spliced  and  cemented. 

3.  The  most  economical  belt  speed  is  from  4000  to 
4500  ft.  per  min. 

4.  The  working  tension  of  a  double  belt  should  not 
exceed  35  Ib.  per  inch  of  width,  but  the  belt  may  be 
first  tightened  to  about  double  this. 


RULES  FOR  WIDTH  OP  BELTS.  187 

5.  Belts   should  be  cleaned  and  greased  every  six 
months. 

6.  The  best  length  is  from  20  to  25  feet  between 
centers. 

94.  Rules  for  Width  of  Belts.  It  will  be  noticed 
that  Mr.  Taylor  recommends  a  working  tension  only 
TO-  to  ¥V  the  breaking  strength  of  the  belt.  He  justifies 
this  by  saying  that  belts  so  designed  gave  much  less 
trouble  from  stoppage  and  repairs  and  were  conse- 
quently more  economical  than  those  designed  by  the 
ordinary  rules. 

In  the  following  formulas  50  Ib.  per  inch  of  width 
is  allowed  for  double  belts  and  30  Ib.  for  single  belts. 
These  are  suitable  values  for  belts  which  are  not  run- 
ning continuously.  The  formulas  may  be  easily 
changed  for  other  thicknesses  and  for  other  values  of 
CT,. 

Let    HP=  horse  power  transmitted. 

D=  diameter  of  driving  pulley  in  inches. 
-ZV=no.  rev.  per  min.  of  pulley. 

The  moment  of  force  transmitted  by  belt  is 

RD    CT.wD    rr 

~ 


ind  HP      TN      C*w  (112) 

=63025=    126050   ' 

Substituting  the  values  assumed  for  CT^  and  solving 
for  w  : 


Single  belts  «p=420o™  .....  (113) 
Double  belts  w=2500—  4^.  .....  (114) 


188  MACHINE  DESIGN. 

The  most  convenient  rules  for  belting  are  those 
which  give  the  horse-power  of  a  belt  in  terms  of  the 
surface  passing  a  fixed  point  per  minute. 

In  formula  (118) 


we  will  substitute  the  following  : 

W=  width  of  belt  in  feet  =~ 

V  —  velocity  in  ft.  per  min.  =  v-   r  - 

La 


1260507T 


Substituting  values  of  C  and  T^  as  before  and 
solving  for  WV=  square  feet  per  minute  we  have  ap- 
proximately : 

Single  belts   WV=90HP.     .     .     .     ,(115) 
Double  belts  TFF=55HP.     .     .     .     .(116) 

95.  Speed  of  Belting.  As  in  the  case  of  pulley 
rims,  so  in  that  of  belts  a  certain  amount  of  tension  is 
caused  by  the  centrifugal  force  of  the  belt  as  it  passes 
around  the  pulley. 

From  equation  (89)      S=^^ 

where  v=  velocity  in  ft.  per  sec. 

w=  weight  of  material  per  cu.  in. 
S  =  tensile  stress  per  sq.  in. 

To  make  this  formula  more  convenient  for  use  we 
will  make  the  following  changes  in  the  constants  : 


RULES  FOR  WIDTH  OF  BELTS.  189 

Let     V=  velocity  of  belt  in  ft.  per  ramute=60t;. 
w=  weight  of  ordinary  belting. 

=  .032  per  cu.  in. 
S,=  tensile  stress  per  inch  width,  caused  by 

centrifugal  force. 
=  about  T\  8  for  single  belts. 

Then  t,= 


Substituting  these  values  in  (89)  and  solving  for  Sl 

V2 


l      1610000 

The  speed  usually  given  as  a  safe  limit  for  ordinary 
belts  is  3000  ft.  per  min.,  but  belts  are  sometimes  run 
at  a,  speed  exceeding  6000  ft.  per  min. 

Substituting  different  values  of  Fin  the  formula  we 
have  : 

F=3000  S,=  5.591b. 

F=4000  S,=  9.941b. 

F=5000  $  =  15.53  Ib. 

F=6000  $  =  22.3610. 

The  values  of  $  for  double  belts  will  be  nearly 
twice  those  given  above.  At  a  speed  of  5000  ft.  per 
minute  the  maximum  tension  per  inch  of  width  on  a 
single  belt  designed  by  formula  (113),  if  we  call 
C  =  .5,  will  be  : 

(30X2)  +  15.=Y5  Ib. 

giving  a  factor  of  safety  of  eight  or  ten  at  the  splices. 
In  a  similar  manner  we  find  the  maximum  tension 
per  inch  of  width  of  a  double  belt  to  be  : 

=1301b. 


190  MACHINE  DESIGN. 

and  the  margin  of  safety  about  the  same  as  in  single 
belting. 

A  double  belt  is  stiff er  than  a  single  one  and  should 
not  be  used  on  pulleys  less  than  one  foot  in  diameter. 
Triple  belts  can  be  used  successfully  on  pulleys  over  20 
inches  in  diameter. 

96.  Manila  Rope  Transmission.  Eopes  are  some- 
times used  instead  of  flat  belts  for  transmitting  power 
short  distances.  They  possess  the  following  advan- 
tages :  they  are  cheaper  than  belts  in  first  cost ;  they 
are  flexible  in  every  direction  and  can  be  carried 
around  corners  readily.  They  have  however  the  dis- 
advantage of  being  less  efficient  in  transmission  than 
leather  belts  and  less  durable  ;  they  are  also  some- 
what difficult  to  splice  or  repair. 

There  are  two  systems  of  rope  driving  in  common 
use  :  the  English  and  the  American.  In  the  former 
there  are  as  many  separate  ropes  as  there  are  grooves 
in  one  pulley,  each  rope  being  an  endless  loop  always 
running  in  one  groove. 

In  the  American  system  one  continuous  rope  is  used 
passing  back  and  forth  from  one  groove  to  another 
and  finally  returning  to  the  starting  point. 

The  advantage  of  the  English  system  consists  in 
the  fact  that  one  of  the  ropes  may  fail  without  causing 
a  breakdown  of  the  entire  drive,  there  usually  being 
two  or  three  ropes  in  excess  of  the  number  actually 
necessary.  On  the  other  hand  the  American  system 
has  the  advantage  of  a  uniform  regulation  of  the  ten- 
sion on  all  the  plies  of  rope.  The  guide  pulley,  which 
guides  the  last  slack  turn  of  rope  back  to  the  starting 
point,  is  usually  also  a  tension  pulley  and  can  be 
weighted  to  secure  any  desired  tension.  The  English 


STRENGTH  OF  MANILA  ROPES. 

method  is  most  used  for  heavy  drives  from  engines  to 
head  shafts  ;  the  American  for  lighter  work  in  dis- 
tributing power  to  the  different  rooms  of  a  factory. 
The  grooves  in  the  pulleys  for  manila  or  cotton  ropes 
usually  have  their  sides  inclined  at  an  angle  of  about 
45°,  thus  wedging  the  rope  in  the  groove. 

The  Walker  groove  has  curved  sides  as  shown  in 
Fig.  83,  the  curvature 
increasing  towards  the 
bottom.  As  the  rope 
wears  and  stretches  it 
becomes  smaller  and  sinks 
deeper  in  the  groove  ;  the 
sides  of  the  groove  being 
more  oblique  near  the 
bottom,  the  older  rope  is 

not  pinched  so  hard  as  the    '  ~ — "~ ' 

newer  and  this  tends  to 

throw  more  of  the  work  on  the  latter. 

97.  Strength  of  Manila  Ropes.  The  formulas  for 
transmission  by  ropes  are  similar  to  those  for  belts, 
the  values  for  $  and  <f>  being  changed.  The  ultimate 
tensile  strength  of  manila  and  hemp  rope  is  about 
10000  Ib.  per  sq.  in. 

To  insure  durability  and  efficiency  it  has  been  found 
best  in  practice  to  use  a  large  factor  of  safety.  Prof. 
Forrest  E.  Jones  in  his  book  on  Machine  Design 
recommends  a  maximum  tension  of  200  dz  pounds 
where  d  is  the  diameter  of  rope  in  inches.  This  cor- 
responds to  a  tensile  stress  of  255  Ib.  per  sq.  in.  or  a 
factor  of  safety  of  about  40. 

The  value  of  /  for  manila  on  metal  is  about  0.12, 
but  as  the  normal  pressure  between  the  two  surfaces 


192  MACHINE  DESIGN. 

is  increased  by  the  wedge  action  of  the  rope  in  the 
groove  we  shall  have  the  apparent  value  of/: 

/1= /-s-sin-^-  where 

A 

a. = angle  of  groove, 
For  a=4r50  to  30° 

fl  varies  from  0.3  to  0.5  and  these  values  should  IK 
used  in  formula  (110). 

(1— e~*  )  in  this  formula,  for  an  arc  of  contact  of 
150°,  becomes  either  .54  or  .73  according  as/1  is  taken 
0.3  or  0.5. 

If  Tl  is  assumed  as  250  Ib.  per  sq.  in.,  the  force  R 
transmitted  by  the  rope  varies  from  135  Ib.  to  185 
Ib.  per  sq.  in.  area  of  rope  section. 

The  following  table  gives  the  horse-power  of  manila 
ropes  based  on  a  maximum  tension  of  255  Ib.  per  sq.  in. 

TABLE  XXXIII. 

Table  of  the  horse-power  of  transmission  rope,  reprinted 
from  the  transactions  of  the  American  Society  of  Mechanical 
Engineers,  Vol.  12,  page  230,  Article  on  "Rope  Driving"  by 
C.  W.  Hunt. 

The  working  strain  is  800  Ib.  for  a  2-inch  diameter  rope  and 
is  the  same  at  all  speeds,  due  allowance  having  been  made  for 
loss  by  centrifugal  force. 


WIRE  ROPE  TRANSMISSION. 


193 


Diameter-  II 
Rope, 
Inches.  || 

SPEED  OF  THE  ROPE  IN   FEET  PER  MINUTE. 

1st 

1500 

2000 

2500 

3000 

3500 

4000 

4500 

5000 

6000 

7000 

1 

3.3 

4.3 

5.2 

5.8 

6.7 

7.2 

7.7 

7.7 

7.1 

4.9 

30 

7 

4.5 

5.9 

7.0 

8.2 

9.1 

9.8 

10.8 

10.8 

9.3 

6.9 

36 

1 

5.8 

7.7 

9.2 

10.7 

11.9 

12.8 

13.6 

13.7 

12.5 

8.8 

42 

It 

9.2 

12.1 

14.3 

16.8 

18.6 

20.0 

21.2 

21.4 

19.5 

13.8 

54 

1* 

13.1 

17.4 

20.7 

23.1 

26.8 

28.8 

30.6 

30.8 

28.2 

19.8 

60 

If 

18.0 

23.7 

28.2 

32.8 

36.4 

39.2 

41.5 

41.8 

37.4 

27.6 

72 

2 

23.1 

30.8 

36.8 

42.8 

47.6 

51.2 

54.4 

54.8 

50.0 

35.2 

84 

98.  Wire  Rope  Transmission.  Wire  ropes  have 
been-  used  to  transmit  power  where  the  distances  were 
too  great  for  belting  or  hemp  rope  transmission.  The 
increased  use  of  electrical  transmission  is  gradually 
crowding  out  this  latter  form  of  rope  driving. 

For  comparatively  short  distances 
of  from  100  to  500  yards  wire  rope  still 
offers  a  cheap  and  simple  means  of 
carrying  power. 

The  pulleys  or  wheels  are  entirely 
different  from  those  used  with  manila 
ropes. 

Fig.  84  shows  a  section  of  the  rim 
of  such  a  pulley.  The  rope  does  not 
touch  the  sides  of  the  groove  but 
rests  on  a  shallow  depression  in  a 


Fig.  84. 


wooden,  leather  or  rubber  filling  at  the  bottom.     The 
high  side  flanges  prevent  the  rope  from  leaving  the 
pulley  when  swaying  on  account  of  the  high  speed. 
The  pulleys  must  be  large,  usually  about  100  times 
'3 


MACHINE  DESIGN. 


the  diameter  of  rope  used,  and  run  at  comparatively 
high  speeds.  The  ropes  should  not  be  less  than  200 
feet  long  unless  some  form  of  tightening  pulley  is  used. 
—Table  XXXIV.  is  taken  from  Roebling. 

Long  ropes  should  be  supported  by  idle  pulleys  every 
400  feet. 

TABLE  XXXIV. 

TRANSMISSION   OF  POWER  BY  WIRE  ROPE. 

Showing  necessary  size  and  speed  of  wheels  and  rope  to  obtain 
any  desired  amount  of  power. 


«M 

a 

~ 

t|~l  -iJ 

03 

; 

!_, 

^  0 

LJ 

i 

—  _ 

0  § 

;_, 

> 

Diamete] 
Wheel  in 

Number 
Revoluti 

Diamete 
of  Rope. 

f 

Diamete 
Wheel  in 

Number 
Revoluti 

-2    13 

I  1 

S  "8 

o 

w 

4 

80 

5-8 

3.3 

10 

80 

11-16 

58.4 

100 

5-8 

4.1 

100 

11-16 

73. 

120 

5-8 

5. 

120 

11-16 

87.6 

140 

5.8 

5.8 

140 

11-16 

102.2 

5 

80 

7-16 

6.9 

11 

80 

11-16 

75.5 

100 

7-16 

8.6 

100 

11-16 

94.4 

120 

7-16 

10.3 

120 

11-16 

113.3 

140 

7-16 

12.1 

140 

11-16 

132.1 

6 

80 

1—2 

10.7 

.12 

80 

3-4 

99.3 

100 

1-2 

13.4 

100 

3-4 

124.1 

120 

1-2 

16.1 

120 

3-4 

148.9 

140 

1-2 

18.7 

140 

3-4 

173.7 

7 

80 

9-16 

16.9 

13 

80 

3-4 

122.6 

100 

9-16 

21.1 

100 

3-4 

153.2 

120 

9-16 

25.3 

120 

3-4 

183.9 

8 

80 

5-8 

22. 

14 

80 

7-8 

148. 

100 

5-8 

27.5 

100 

7-8 

185. 

120 

5-8 

33.0 

120 

7-8 

222. 

9 

80 

5-8 

41.5 

15 

80 

7-8 

217. 

100 

5-8 

51.9 

100 

7-8 

259. 

120 

5-8 

62.2 

120 

7-8 

300. 

WIRE  ROPE  TRANSMISSION.  195 

PKOBLEMS. 

1.  Design  a  main  driving  belt  to  transmit  150  HP.  from  a 
belt  wheel  18  ft.  in  diameter  and  making  80  rev.  per  min. 
The  belt  to  be  double  leather  without  rivets. 

2.  Investigate  driving  belt  on  Allis  engine  and  calculate  the 
horse-power  it  is  capable  of  transmitting  economically. 

3.  Calculate  the  total  maximum  tension  per  inch  of  width 
due  to  load  and  to  centrifugal  force  of  the  driving  belt  on  the 
motor  used  for  driving  machine  shop,  assuming  the  maximum 
load  to  be  10  HP. 

4.  Design  a  manila  rope  drive,  English  system,  to  transmit 
500  HP. ,  the  wheel  on  the  engine  being  20  feet,  in  diameter 
and  making  60  rev.  per  min.     Use  Hunt's  table  and  then 
check  by  calculating  the  centrifugal  tension  and  the  total 

u2 
maximum  tension  on  each  rope.    Assume  S=  ^    where    v= 

feet  per  second. 

5.  Design  a  wire  rope  transmission  to  carry  120  HP.  a  dis- 
tance of  one-quarter  mile  using  two  ropes.     Determine  working 
and  maximum  tension  on  rope,  length  of  rope,  diameter  and 
speed  of  pulleys  and  number  of  supporting  pulleys. 


INDEX. 


ART.  PAGE. 

ABBREVIATIONS 2  1 

ADJUSTMENT  OF  BEARINGS 43  96 

ALLOYS 3  4 

ARMS  OF  PULLEYS 77  157 

BALL  BEARINGS, 

In  general 57  118 

Conical 58  119 

Cylindrical 58  118 

Design  of 61  123 

Materials  of 60  122 

Step  or  thrust 59  120 

BEAMS,  formulas  for 5  12 

Of  uniform  strength 6  12 

BEARINGS, 

Adjustment  of 43  96 

Ball 57  118 

Cylindrical... 42  96 

Engine... , 43  97 

Lathe ......:...  43  98 

Lubrication  of 44  99 

Roller , 62  123 

Sliding 36  86 

Steporthrust 51  111 

Thrust 56  116 

BELTING, 

Centrifugal  tension  of ...:. 95  189 

Friction  of .V 91  184 

Speed  of 95  188 

Strength  of 92  186 

Taylor's  experiments  on. 93  186 

Width  of 94  187 

BOILER  SHELLS ...  11  25 

Tubes 14  38 

197 


198  MACHINE  DESIGN. 

ART.  PAGE. 

BOLTS, 

Coupling 69  138 

Dimensions  of 18  54 

Eyeorhook 20  57 

BRONZES 3  5 

BUTT  JOINTS 23  62 

CABINET  SUPPORTS 9  19 

CAPS  AND  BOLTS 50  109 

CAST  IRON 3  3 

CENTRIFUGAL  OILERS 44  101 

CHAIN  DRIVING, 

Block 78  159 

Roller 78  160 

Silent 79  161 

CLUTCHES, 

Conical 68  137 

Roller ' 68  138 

Weston 68  136 

COLLAR  BEARING 53  113 

Compound 56  116 

COLUMN  FORMULAS 5  9 

CONSTANTS, 

Columns 5  10 

Cross-sections ,..  5  11 

COTTERS 29  69 

COUPLINGS, 

Bolts  for 69  138 

Clutch 68  135 

Flange 67  133 

.   Muff 67  135 

Sleeve 67  134 

CRANKS 80  163 

CRANK  PINS, 

Heatingof 47  104 

Pressure  on 46  103 

CROSS-SECTIONS 10  21 

CYLINDERS, 

Hydraulic 12  27 

Steam 16  42 

DESIGN,  GENERAL  PRINCIPLES  OP 8  15 

Discs,  ROTATING , .  86  177 


INDEX.  199 

ART.  PAGE. 

Disc,— Continued. 

Conical 88  180 

Logarithmic 89  181 

Plain 87  178 

Speeds  of 90  181 

FACTORS  OF  SAFETY 7  13 

FLAT  PLATES, 

Formulas  for 17  48 

Tests  of 17  51 

FLYWHEELS 81  166 

Experiments  on 83  170 

Rirn  Joints  of 83  175 

Safe  Speed  of 82  167 

FLUES,  STRENGTH  OF. 14  38 

FORMULAS,  GENERAL 5  8 

FRAME  DESIGN 10  20 

FRICTION, 

Belts 91  184 

Journals 45  102 

Experiments  on 48  107 

Pivots 52  112 

Schiele  pivot 55  114 

GEARS, 

Arms  of 77  157 

Rims  of , 77  157 

Teethof 72  146 

GIBS 28  72 

"    38  89 

GUIDES,  CIRCULAR 40  91 

HANGERS 71  141 

HEATING  OF  JOURNALS 47  107 

HOOKS,  DESIGN  OF 20  57 

IRON, 

Cast t 3  3 

Malleable 3  4 

Wrought 3  2 

JOINTS, 

Butt 23  62 

Lap 22  61 

Riveted. 21  58 

JOINT  PINS.  .  28  69 


200  MACHINE  DESIGN. 

ART.  PAGE. 

JOURNALS. 42  96 

Experiments  on 48  107 

Friction  of 45  102 

Heating  of 47  104 

Pressure  on 46  103 

Strength  of , 49  108 

KEYS, 

Cotter 28  69 

Shafting 70  139 

Woodruff 70  141 

LAP  JOINTS 22  61 

LEGS  OF  MACHINES 9  19 

LEVERS 80  163 

LUBRICATION  OF  BEARINGS 44  99 

MACHINE  FRAMES 10  20 

MALLEABLE  IRON  3  4 

MATERIALS  OF  CONSTRUCTION 3  2 

NOTATION 4  8 

OIL  CUPS , 44  100 

PACKINGS  FOR  GLANDS 41  92 

PIPE, 

Sizes 13  30 

Fittings ,..15  40 

PIVOTS, 

Conical .... 54  113 

Flat ... 52  112. 

Schiele 55  114 

PLATES, 

Flat 17  48 

Narrow 27  68 

PRESSURE  ON  JOURNALS 46  103 

PULLEYS, 

Arms  of 77  157 

Cast  Iron ,, 83  175 

Wooden 84  176 

RING  OILER. . ...  44  101 

RIVETED  JOINTS 21  58 

Efficiency  of 24  63 

Narrow  Plates 27  68 

Practical  rules  for 26  fit 

Special  forms  of 25  63 


INDEX.  201 

ART.  PAGE. 

ROLLER  BEARINGS, 

Conical 63  124 

Cylindrical 62  123 

Hyatt.... 64  125 

Step.... 65  127 

Tests  of 64  126 

ROPE,  TRANSMISSION, 

Manila...:..... 96  190 

Strength  of 97  191 

Wire....:. 98  193 

SCHIELE  PIVOT 55  114 

SCREWS,  MACHINE 19  57 

SHAFTING, 

Diameter  of 66  132 

Keys  for 70  139 

Span  of 66  133 

Strength  of 66  130 

SHELLS,  STRENGTH  OF, 

Thick 12  27 

Thin 11  25 

SLIDES, 

Angular 37  87 

Flat 39  89 

Gibbed 38  88 

SPRINGS, 

Elliptic 35  83 

Flat 34  81 

Helical 30  73 

Square  wire 31  75 

Testsof f 32  76 

Torsion „ 33  79 

STEAM  CYLINDERS, 

Strength  of 16  42 

Testsof 16  45 

STEEL 3  3 

STRENGTH  OF  METALS, 

Cast 3  7 

Wrought t 3  6 

STUFFING  BOXES 41  91 

SUPPORTS}  MACHINE 9  is 


202  MACHINE  DESIGN. 

ART.  PAftE. 

TEETH  OF  GEARS, 

Bevel 76  155 

Cut 73  150 

Experiments  on 75  153 

Lewis'  formula  for 74  151 

Proportions  of 72  143 

Strength  of 73  148 

Velocity  of 74  153 

THRUST  BEARING 56  116 

TUBES,  BOILER 14  33 

UNITS  AND  DEFINITIONS 1  i 

WROUGHT  IRON 3  g 


Hall's  College  Laboratory  Manual  of  Physics 

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